# **SpringerBriefs in Mathematics Gunther Cornelissen · Norbert Peyerimhof**

# **Twisted Isospectrality, Homological Wideness, and Isometry**  A Sample of Algebraic Methods in Isospectrality

# **SpringerBriefs in Mathematics**

### **Series Editors**

Nicola Bellomo, Torino, Italy Michele Benzi, Pisa, Italy Palle Jorgensen, Iowa, USA Tatsien Li, Shanghai, China Roderick Melnik, Waterloo, Canada Otmar Scherzer, Linz, Austria Benjamin Steinberg, New York, USA Lothar Reichel, Kent, USA Yuri Tschinkel, New York, USA George Yin, Detroit, USA Ping Zhang, Kalamazoo, USA

SpringerBriefs present concise summaries of cutting-edge research and practical applications across a wide spectrum of fields. Featuring compact volumes of 50 to 125 pages, the series covers a range of content from professional to academic. Briefs are characterized by fast, global electronic dissemination, standard publishing contracts, standardized manuscript preparation and formatting guidelines, and expedited production schedules.

### **Typical topics might include:**


**SpringerBriefs in Mathematics** showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. All works are peer-reviewed to meet the highest standards of scientific literature.

*Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews, and zbMATH.*

Gunther Cornelissen • Norbert Peyerimhoff

# Twisted Isospectrality, Homological Wideness, and Isometry

A Sample of Algebraic Methods in Isospectrality

Gunther Cornelissen Mathematisch Instituut Universiteit Utrecht Utrecht, The Netherlands Norbert Peyerimhoff Science Laboratories Durham Univ, Dept of Math Sci Durham, UK

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-031-27703-0 ISBN 978-3-031-27704-7 (eBook) https://doi.org/10.1007/978-3-031-27704-7

This work was supported by Nederlandse Organisatie voor Wetenschappelijk Onderzoek

© The Author(s) 2023. This book is an open access publication.

**Open Access** This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this book are included in the book's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Open Access publication made possible by the Dutch Research Council (NWO) in connection with project "Utrecht Geometry Centre" (project number 022.006.004; research programme "Graduate Schools").

# **Preface**

The problems we deal with in this book date back at least to the Wolfkehl lectures *Alte und Neue Frage der Physik* delivered in Göttingen in 1910 by Hendrik A. Lorentz [63], who asked for the relation between spectra of operators (providing physically observable quantities, such as radiation) and geometric properties of the observed objects; more precisely, he asked—in modern terminology—whether for a geometric shape, the eigenvalues of the Laplace operator, with suitable boundary conditions, determine the volume of the shape, adding that "for several simple shapes where the computations can be done, this will be confirmed in a Leiden thesis".1 That was the thesis of Johanna Reudler [82], published in 1912, done concurrently with Hermann Weyl's 1911 proof for the general case, using the theory of integral equations [100].

An even bolder but natural follow-up question is whether a geometric shape is completely determined by the eigenvalues.2 Said more succinctly, "Does isospectrality imply isometry?", or—in the catchy formulation of Bers that was eternalised as title of the famous 1966 article by Mark Kac—"Can one hear the shape of a drum?". As the original question for planar domains with Dirichlet boundary conditions remained out of reach, the problem was considered for the spectrum of the Laplace-Beltrami operator on closed smooth oriented Riemannian manifolds. In that setting, the question was answered in the negative, first by John Milnor's construction of two isospectral, non-isometric tori of dimension 16 [71], then by Marie-France Vignéras' construction of such compact Riemann surfaces [97] and then by a general construction of Toshikazu Sunada [90]. The unifying principle behind these constructions is that they are *number-theoretical*, being based, respec-

<sup>1</sup> In [55], this is translated as "has been confirmed", complicating the search for the further unnamed thesis.

<sup>2</sup> Already in 1881, Arthur Schuster had pronounced that "It would baffle the most skilful mathematician to solve the inverse problem and to find out the shape of a bell by means of the sounds which it is capable of sending out. And this is the problem which ultimately spectroscopy hopes to solve in the case of light. In the meantime we must welcome with delight even the smallest step in the desired direction" [85].

tively, on the arithmetic theory of quadratic forms, maximal orders in quaternion algebras and the theory of arithmetic equivalence. (As an aside, Sunada's result turned out to be a crucial ingredient in the final solution of the original question about planar domains in the negative by Gordon et al. [43]; see [26] for a popular account). This is the starting point for an approach to the problem of isospectrality that is rooted more in algebra (group theory, representation theory and number theory) than in analysis.

This book focusses on explaining and advancing that algebraic approach. The setup is the same as in Sunada's work: we are given a manifold (or, more generally, a developable orbifold) *M*<sup>0</sup> and two closed Riemannian manifolds *M*<sup>1</sup> and *M*<sup>2</sup> with a finite covering map to *M*0. We find a *G*-Galois cover *M* of *M*<sup>0</sup> for some finite group *G*, such that *M* is a common finite cover of *M*<sup>1</sup> and *M*2, corresponding to two subgroups *H*<sup>1</sup> and *H*<sup>2</sup> of *G*, respectively. Sunada shows that if *H*<sup>1</sup> and *H*<sup>2</sup> are weakly conjugate in *G* (meaning that the group representations of *G* corresponding to the permutation action of *G* on the cosets of *H*<sup>1</sup> and *H*2, respectively, are isomorphic), then *M*<sup>1</sup> and *M*<sup>2</sup> are isospectral, but not necessarily isometric, or even covering equivalent (meaning that *H*<sup>1</sup> and *H*<sup>2</sup> are conjugate subgroups of *G*).

A new feature of this work is that we give a spectral characterisation of when *M*<sup>1</sup> and *M*<sup>2</sup> are equivalent Riemannian covers (in particular, isometric), assuming a representation-theoretic condition of "homological wideness", that involves the action of *G* on the first homology group of *M*. The condition holds, for example, when there exists a rational homology class on *M* whose orbit under *G* consists of |*G*| linearly independent homology classes. We prove that, under this condition, Riemannian covering equivalence is the same as isospectrality of finitely many twisted Laplacians on the manifolds, acting on sections of flat bundles corresponding to specific representations of the fundamental groups of the manifolds involved. Using the same methods, we provide spectral criteria for weak conjugacy and strong isospectrality. In the negative curvature case, we formulate an analogue of our result for the length spectrum. We study examples where the representation theoretic condition does and does not hold. For example, when *M*<sup>1</sup> and *M*<sup>2</sup> are commensurable non-arithmetic closed Riemann surfaces of negative Euler characteristic, there is always such an *M*0, and the condition of homological wideness always holds.

As indicated above, the proofs are inspired by number-theoretical analogues3 and use representation theory, and are thus firmly rooted in the tradition of the work of Toshikazu Sunada as well as Hubert Pesce. The notion of homological wideness might be of independent interest to topologists.

The methods combine concepts and tools from different fields, and we have been rather discursive on details, background and recaps. We hope the book also

<sup>3</sup> Ironically, the Wolfkehl lectures were funded by an endowment that was available as long as Fermat's Last Theorem had *not* been solved, so in a sense, we owe both the origin as well as the solution of the isospectrality problem to number theory, albeit in one case due to our prolonged inability to solve a number theoretical problem.

serves as an introduction to basic concepts and results in the algebraic approach to isospectrality, namely: around fiber products of orbifolds, twisted Laplacians, zeta functions and spectra of operators, strong isospectrality, *G*-sets, weak and strong conjugacy and isospectrality, monomial representations, wreath product realisations, homology groups in Galois covers and homology representations. We have also made sure the text is full of examples, including an entire chapter devoted to examples and counterexamples. At the end of most chapters is a short list of open problems or possible future projects.

The introduction gives a technical description of the new results. After the introduction follows a *Leitfaden*, where we list some of the more folklore (but difficult to find) results to be found in the current text.

It is our pleasure to thank Durham University and Utrecht University for their hospitality. We thank Jens Funke for sparking this project by bringing us together at the 2017 BMC. At various points, we received help, triggers and feedback from Alex Bartel, Stefan Bechtluft-Sachs, Mikolaj Fr ˛aczyk, Nigel Higson, Gerhard Knieper, Matthias Lesch, Juan Pablo Rossetti, Jeroen Sijsling and Harry Smit and we thank all of them wholeheartedly. We also thank the reviewers for their feedback on an earlier version.

Utrecht, The Netherlands Gunther Cornelissen Durham, UK Norbert Peyerimhoff June 2022

# **Contents**



# **List of Symbols**

### **General**


## **Manifolds**


### **Groups**



# **Representations**


### **Operators**


*L*2*(M, E)* Completion of *C*∞*(M, E)* on a closed Riemannian manifold *M* w.r.t. the inner product corresponding to the volume form

*σM(A)* Spectrum of A, i.e., the multiset of eigenvalues in *L*2*(M, E)* of a symmetric second order elliptic differential operator *A* acting on *C*∞*(M, E)* for some Hermitian bundle *E* with non-negative eigenvalues; also denoted *σ (A)* if the manifold is fixed


# **Chapter 1 Introduction**

The chapter contains an overview of the main new results in the book: a spectral characterisation of covering equivalence of Riemannian manifolds using spectra of finitely many twisted Laplacians, a more detailed version for non-arithmetic Riemann surfaces, a version using the length spectrum. It also contains a spectral characterisation of weak conjugacy. The chapter contains pointers forward to other chapters that contain an exposition of the basic tools and methods, as well as detailed constructions, proofs and examples.

### **1.1 Setup and Conditions**

Let *M*<sup>1</sup> and *M*<sup>2</sup> be a pair of connected closed oriented smooth Riemannian manifolds. There exist such *M*<sup>1</sup> and *M*<sup>2</sup> that are not isometric, but *isospectral*, i.e., they have the same Laplace spectrum with multiplicities. This leaves open the question whether equality of spectra of other geometrically defined operators on *M*<sup>1</sup> and *M*<sup>2</sup> is equivalent to *M*<sup>1</sup> and *M*<sup>2</sup> being isometric. In this text we investigate the use of twisted Laplacians (acting on sections of flat bundles constructed from representations of fundamental groups) in answering this question. We will see that under two conditions on *M*<sup>1</sup> and *M*2, equality of finitely many suitably twisted Laplace spectra implies isometry of the manifolds. That at least some condition is necessary for such a result to hold is illustrated by the existence of simply connected isospectral non-isometric manifolds [87].

The **first condition** is the following: we suppose that the manifolds *M*<sup>1</sup> and *M*<sup>2</sup> are finite Riemannian coverings of a developable Riemannian orbifold *M*<sup>0</sup> (meaning that the universal covering of *M*<sup>0</sup> is a manifold), expressed through a diagram

Such a diagram may be extended to a diagram of finite coverings

where *M* is a connected closed smooth Riemannian manifold *M* with *q*<sup>1</sup> : *M* - *M*<sup>1</sup> := *H*1\*M*, *q*<sup>2</sup> : *M* - *M*<sup>2</sup> := *H*2\*M* and *q* : *M* - *M*<sup>0</sup> := *G*\*M* Galois covers (Proposition 2.4.1); in particular, *M*<sup>1</sup> and *M*<sup>2</sup> are commensurable. Commensurability is in general weaker than the existence of a diagram (1.1) (see Proposition 2.5.4). However, if *M*<sup>1</sup> and *M*<sup>2</sup> are commensurable hyperbolic manifolds, then a diagram (1.1) (with an orbifold *M*0) exists if the corresponding lattices are not arithmetic (Proposition 2.5.2).

The **second condition** is related to the action of *G* on the first homology group of *M*. Let **F** denote the field with  elements. In terms of the data in diagram (1.2), we require that for some prime number  not dividing |*G*|, the **F***-*[*G*]-module H1*(M,* **F***-)* = H1*(M)* ⊗**<sup>Z</sup> F** contains the permutation representation of *G* acting by left multiplication on the cosets *G/Hi* for either *i* = 1 or *i* = 2 (or both), i.e., *(*Ind*<sup>G</sup> Hi* **1***)*⊗**<sup>Z</sup> F** is an **F***-*[*G*]-submodule of H1*(M,* **F***-)*. Concretely, this means that if there are *n* cosets, then the **F***-*-vector space H1*(M,* **F***-)* contains *n* linear independent vectors that are permuted in the same way as those *n* cosets under the action of any *g* ∈ *G*.

This second condition is implied by the stronger requirement that the *G*-action is **F***-*-*homologically wide*, where, for a general field *K*, we say that the *G*-action is *K*-*homologically wide* if the regular representation *K*[*G*] of *G* occurs in the homology representation *G* → Aut*(*H1*(M, K))* (Lemma 8.1.2). This condition has the advantage of no longer referring to *M*<sup>1</sup> and *M*<sup>2</sup> (or *H*<sup>1</sup> and *H*2).

An even stronger, geometrically tangible, condition is that the action of *G* on *M* is **Q**-homologically wide; this means that there is a free homology class *ω* on *M* such that the elements {*gω*: *g* ∈ *G*} are linearly independent in H1*(M,* **Q***)*. This is diametric to the condition of homologically trivial group actions found more frequently in the literature (see, e.g., [99]). A **Q**-homologically wide action is **F***-* homologically wide for any coprime to |*G*| (Lemma 8.2.1).

In Chap. 9, we discuss **Q**-homological wideness for certain low dimensional manifolds and locally symmetric spaces. For non-trivial fixed-point free group actions on orientable surfaces, **Q**-homologically wideness is equivalent to any of the spaces *M,M*0*, M*<sup>1</sup> or *M*<sup>2</sup> having negative Euler characteristic (Proposition 9.1.1). A (not necessarily fixed point free) holomorphic group action on a Riemann surface *M* is **Q**-homologically wide if the Euler characteristic of the quotient surface *M*<sup>0</sup> satisfies *χM*<sup>0</sup> *<* 0 (Proposition 9.1.2). In higher dimension, the picture can vary widely: for any *n* ≥ 3, we use standard surgery methods to construct an *n*-manifold with a free action by any given non-trivial group that is or is not **Q**-homologically wide (Proposition 9.3.1 and Corollary 9.3.3). Since locally symmetric spaces of rank ≥ 2 have trivial rational homology, no non-trivial group action on them can be **Q**-homologically wide (see Sect. 9.4). On the other hand, in rank one, the condition relates to decomposition results for automorphic representations (see Sect. 9.5). A disadvantage of **Q**-homological wideness is that it discards torsion homology; in Sect. 9.7 we give an example of some non-trivial **F**5-homologically wide group actions on the Seifert–Weber dodecahedral space (that has homology H1*(M,***Z***)* ∼= *(***Z***/*5**Z***)* 3).

### **1.2 Overview of Main Results**

Our results involve spectra of *twisted Laplace operators ρ* corresponding to unitary representations *ρ* : *π*1*(M)* → U*(N ,* **C***)*; these are symmetric second order elliptic differential operators acting on sections of flat bundles *Eρ* over *M*; the sections are conveniently described as smooth vector valued functions on the universal cover that are equivariant with respect to the representation, and on these, *ρ* acts (componentwise) like the usual Laplacian of the universal cover (cf. Chap. 3). In fact, our representations will factor through a specific finite group, allowing for a very concrete description of the operators (cf. Sects. 3.6 and 3.7). Denote the spectrum of such an operator by *σM(ρ)*, where the index *M* indicates that the Laplacian is defined on sections over the space *M*.

Returning to the setup in diagram (1.1), we call *M*<sup>1</sup> and *M*<sup>2</sup> *equivalent Riemannian covers of M*<sup>0</sup> if there is an isometry between them induced by a conjugacy of the fundamental groups of *M*<sup>1</sup> and *M*<sup>2</sup> inside that of *M*<sup>0</sup> (since *M*<sup>0</sup> is developable, its universal covering is a manifold and we mean the subgroup of its isometry group that fixes *M*<sup>0</sup> pointwise, see Lemma 2.1.2). Our main result states that in this situation isometry can be detected via the spectra of finitely many specific twisted Laplacians.

**Theorem 1.2.1** *Suppose we have a diagram* (1.1) *and suppose that the action of G on M in the extended diagram* (1.2) *is* **F***--homologically wide for some prime coprime to* |*G*|*. Then M*<sup>1</sup> *and M*<sup>2</sup> *are equivalent Riemannian covers of M*<sup>0</sup> (*in particular, isometric*) *if and only if the multiplicity of zero in the spectra of a finite number of specific twisted Laplacians on M*<sup>1</sup> *and M*<sup>2</sup> *is equal. In fact, at most*  2*-*|Hom*(H*2*,* **C**∗*)*| *equalities suffice.* 

For a precise formulation of the required twists and a more technical condition (denoted *(*∗*)*) that is weaker than homological wideness, the reader is encouraged to glance at Theorem 6.4.1, which is a more detailed formulation of Theorem 1.2.1, our main result. The detailed formulation reveals that the representations occurring in the theorem are constructed explicitly using induction and restriction of special characters (termed "solitary" below) on groups corresponding to a specific finite Riemannian covering of *M*, whose existence is guaranteed by the assumption of homological wideness (or the weaker requirement *(*∗*)*). Riemannian equivalence over *M*<sup>0</sup> is the same as conjugacy of *H*<sup>1</sup> and *H*<sup>2</sup> in *G*; a merit of the theorem is to show that this can be verified via a spectral geometric criterion using twists, where the twists on *M*<sup>1</sup> are constructed using information from *M*<sup>2</sup> and vice versa.

Sunada [90] showed that if we are given a diagram of the form (1.2), and *H*<sup>1</sup> and *H*<sup>2</sup> are *weakly conjugate* (meaning that the permutation representations given by the action of *G* by left multiplication on the cosets *G/H*<sup>1</sup> and *G/H*<sup>2</sup> are isomorphic), *M*<sup>1</sup> and *M*<sup>2</sup> are isospectral for the Laplace operator. The following example illustrates our theorem in such a situation.

*Example 1.2.2* We provide the explicit data for what is maybe the oldest example of weakly conjugate subgroups, due to Gaßmann [38]:

$$G = S\_6, \ H\_1 = \langle (1\mathfrak{I})(34), (1\mathfrak{I})(24) \rangle, \ H\_2 = \langle (1\mathfrak{I})(34), (1\mathfrak{I})(56) \rangle,$$

with both *H*<sup>1</sup> and *H*<sup>2</sup> isomorphic to the Klein four-group, but not conjugate inside *S*<sup>6</sup> [38, pp. 674–675]. As in [90, p. 174], choose a compact Riemann surface *M*<sup>0</sup> of genus 2 and a surjective group homomorphism *π*1*(M*0*)* - *G*. This leads to a diagram of the form (1.2) [90, §2], and homological wideness is immediate from Proposition 9.1.1. In this case, *M*<sup>1</sup> and *M*<sup>2</sup> are inequivalent covers of *M*0, but they have the same Laplace spectrum [90, §2]. We can set *-* = 7 in the main theorem, and, with the group *G* having order 720, inequivalence of *M*<sup>1</sup> and *M*<sup>2</sup> may be verified purely spectrally by checking 56 equalities of multiplicities of zero in the spectrum of twisted Laplacians corresponding to representations of dimension 180. ♦

To show the scope of our result, we discuss several more examples in Chap. 11. Our result should be contrasted with [4, Thm. 1.1], where it is shown that large non-arithmetic hyperbolic manifolds *M* admit arbitrary large sets of (strongly) isospectral but pairwise non-isometric finite Riemann coverings.

Our method of proof for Theorem 1.2.1, presented in Chaps. 2–6, is based on a similar construction of Solomatin [89] for algebraic function fields, which in turn is based on number theoretical work of Bart de Smit in [28]. The analogy between number theory and spectral differential geometry was pioneered by Sunada [90] (see also the survey [93]), and the importance of representation theoretical techniques was pointed out early on by Sunada [91] and Pesce [79]. We have given a self-contained presentation, with references to the number theory literature when appropriate. Our construction uses a certain wreath product of *G* with a cyclic group; in Sect. 7.4, we describe a universality property of such wreath products that should make their appearance look less surprising.

We have formulated our results using spectra, but the analogy to number theory becomes most apparent by instead using spectral zeta functions of (twisted) Laplacians; that this is equivalent is explained in Sects. 3.2 and 3.10, pointing to some subtleties concerning the multiplicity of zero in the spectrum of more general operators.

From our earlier brief discussion of the two conditions in the theorem, we find the following specific result in dimension two.

**Corollary 1.2.3** *Let M*1*, M*<sup>2</sup> *be two commensurable non-arithmetic closed Riemann surfaces. Then they admit a diagram* (1.1) *and, assuming the corresponding orbifold M*<sup>0</sup> *satisfies χM*<sup>0</sup> *<* 0*, isometry of M*<sup>1</sup> *and M*<sup>2</sup> *can be checked by computing the multiplicity of zero in at most* 4*((χM*1*χM*<sup>2</sup> */(χ*orb *<sup>M</sup>*<sup>0</sup> *)*2*)*!*)*<sup>2</sup> *twisted Laplace spectra, where χ*orb *<sup>M</sup>*<sup>0</sup> *is the orbifold Euler characteristic, defined in* (9.1)*.* 

The detailed formulation and proof can be found in Corollary 9.1.4. If one believes in a positive answer to the (open since several decades) question whether commensurability of such Riemann surfaces is implied by their isospectrality, then a purely spectral formulation of the corollary is possible, replacing "commensurable" by "isospectral". Intriguingly, the only cases where a positive answer to the open question is known are arithmetic [81], precisely the ones excluded in the corollary.

In Chap. 12, we study the analogue of Theorem 1.2.1 for the *length spectrum*  on negatively curved manifolds. The statement about agreement of multiplicities of zero in certain spectra is changed into equality of the pole order of certain *L*-series (details are found in Theorem 12.2.4).

**Theorem 1.2.4** *Suppose we have a diagram* (1.1) *of negatively curved Riemannian manifolds with (common) volume entropy h, and suppose that the action of G on M in the extended diagram* (1.2) *is* **F***--homologically wide for some coprime to* |*G*|*. Then M*<sup>1</sup> *and M*<sup>2</sup> *are equivalent Riemannian covers of M*<sup>0</sup> *if and only if the pole order at s* = *h of a finite number of specific L-series of representations on M*<sup>1</sup> *and M*<sup>2</sup> *is equal.* 

Sunada's result [90] quoted above says that weak conjugacy implies isospectrality, but the converse is not necessarily true; this leaves open the question to characterise weak conjugacy of *H*<sup>1</sup> and *H*<sup>2</sup> in *G* in a spectral way using the associated manifolds *M*<sup>1</sup> and *M*2. One of our intermediate results answers this question using induction and restriction from the trivial representation **1**.

**Proposition 1.2.5** *If we have a diagram* (1.2)*, then H*<sup>1</sup> *and H*<sup>2</sup> *are weakly conjugate if and only if the multiplicity of the zero eigenvalue in σMi(*Res*<sup>G</sup> Hi* Ind*<sup>G</sup> Hj* **1***) is independent of i, j* = 1*,* 2*.* 

This result, reformulated in Proposition 4.2.4, is proven by an adaptation of a number theoretical argument of Nagata [73]. The crucial differential geometric ingredient is the spectral characterisation of the multiplicity of the trivial representation in any given representation (Lemma 3.9.1). As a corollary, we get a spectral characterisation of *strong isospectrality* (cf. Definition 4.2.2 and Corollary 4.2.5).

### **Open Problem**

The most pressing question that remains open concerns the case where there is not necessarily a common finite orbifold quotient: are *twisted isospectral* manifolds *M*<sup>1</sup> and *M*<sup>2</sup> (meaning that there is a bijection of all unitary representations of their fundamental groups such that the spectra of the corresponding twisted Laplacians are equal) with *large* fundamental groups (i.e., containing a finite index subgroup with a non-abelian free group as quotient) isometric?

### **Project**

Develop the theory when *M*<sup>1</sup> and *M*<sup>2</sup> are also orbifolds.

### **Project**

Develop an analogous theory for graphs instead of manifolds.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# *Leitfaden*

The *Leitfaden* contains a graph of dependencies between chapters, as well as a list of folklore results that can be found in the main text, of possible independent interest.

### **Interdependency Graph**

The dependencies between chapters is indicated in the graph below. In boldface is the minimal path to reach the main results (Theorems 1.2.1 = 6.4.1 and Theorem 1.2.4 = 12.2.4), in regular font a geometric instead of purely grouptheoretical construction, and in boxed font material on homological wideness and examples. A dashed arrow represents a weak dependence.

### **"Folklore" Results with Forward Pointers**

Apart from the main results, we have included several "folklore" results that are hard to find in the literature. We list some of these.


# **Chapter 2 Manifold and Orbifold Constructions**

In this chapter, we collect some information on various constructions of manifolds, orbifolds, and their covers. Notably, we discuss the notions of fiber product (in the sense of Thurston) and compositum of manifolds over a common developable orbifold, the difference with the set-theoretic fiber product, the connected components of the fiber product, compositum of Galois covers, normal closure of a cover, and the relation between commensurability, arithmeticity, and the existence of certain covers in relation to Mostow rigidity.

### **2.1 Riemannian Coverings and Their Equivalence**

We fix the following notation for the remaining of the text: we assume that M<sup>1</sup> and M<sup>2</sup> are connected smooth oriented closed (i.e., compact with empty boundary) Riemannian manifolds that are *finite* Riemannian coverings of a developable Riemannian orbifold M0, i.e., there is a diagram

of the form (1.1). Recall that the connected space M<sup>0</sup> being a *developable orbifold*  means that it is the quotient of a connected and simply connected Riemannian manifold M-<sup>0</sup> by a cocompact discrete group of isometries -0, that is the (orbifold) fundamental group of M0. For the theory of (good/developable) orbifolds, see, e.g., [95, Ch. 13] [25] [33, §1]. Given this setup, we can write Mi = i\M-<sup>0</sup> for finite index subgroups <sup>i</sup> of -<sup>0</sup> acting without fixed points on M-0.

*Remark 2.1.1* In dimension 2, there is a relation with the theory of branched coverings of surfaces [78, §3] (compare [95, Thm. 13.3.6]). Consider an orbifold cover M<sup>1</sup> → M<sup>0</sup> of degree n, where M<sup>0</sup> is a closed developable 2-orbifold with r elliptic (cone) points {xi}: the local orbifold structure is that of rotation over an angle 2π/ei. Assume that M<sup>1</sup> is a closed 2-manifold. To such an orbifold cover corresponds a branched degree n cover of the corresponding underlying closed surfaces <sup>1</sup> → <sup>0</sup> branched over the r given points such that xi has exactly n/ei points above it with ramification index ei (in particular, ei divides n). A diagram (1.1) corresponds to a diagram of branched coverings of the corresponding surfaces

where, if pj has degree dj (j = 1, 2), we have the added data of r branch points {xi} on <sup>0</sup> and r integers ei dividing gcd(d1, d2) such that there are exactly dj /ei points above xi in j for j = 1, 2 and i = 1,...,r.

**Lemma 2.1.2** *Suppose we have a diagram of manifolds/orbifolds* 

*of the form* (1.2) *and* -0, -1, -<sup>2</sup> *are as above. Suppose is such that* M = -\M-0*. Then the following are equivalent:* 


$$\begin{array}{c} M\_1 \xrightarrow[p\_1]{\varphi} \begin{array}{c} M\_2 \\ \searrow \\ M\_0 \end{array} \end{array} \Big\vert\_{p\_2} \tag{2.1}$$

*Proof* With Hi = i/ and normal in -0, we have that there exists a g ∈ G satisfying gH1g−<sup>1</sup> <sup>=</sup> <sup>H</sup><sup>2</sup> if and only if for any lift <sup>γ</sup><sup>0</sup> <sup>∈</sup> -<sup>0</sup> of g, γ0-<sup>1</sup>γ <sup>−</sup><sup>1</sup> <sup>0</sup> = -2. This shows the equivalence of (i) and (ii). To show the equivalence of (ii) and (iii), we argue as follows. If gH1g−<sup>1</sup> <sup>=</sup> <sup>H</sup><sup>2</sup> for <sup>g</sup> <sup>∈</sup> <sup>G</sup>, a (finite) group of isometries of M, then <sup>ϕ</sup> : <sup>M</sup><sup>1</sup> <sup>→</sup> <sup>M</sup><sup>2</sup> given by ϕ(H1x) <sup>=</sup> <sup>H</sup>2gx (for <sup>x</sup> <sup>∈</sup> <sup>M</sup>-0) satisfies the requirements of (iii). Conversely, if M<sup>1</sup> and M<sup>2</sup> are equivalent Riemannian covers of M0, then the corresponding subgroups -<sup>1</sup> and -<sup>2</sup> of the orbifold fundamental group -<sup>0</sup> are conjugate.

### **2.2 Compositum**

We set M<sup>00</sup> := -00\M-<sup>0</sup> for -<sup>00</sup> := -<sup>1</sup> ∩ -2. This is a finite connected common Riemannian cover of M<sup>1</sup> and M<sup>2</sup> (finiteness follows since [-<sup>0</sup> : -<sup>1</sup> ∩ -2]≤[-0 : -1]·[-<sup>0</sup> : -<sup>2</sup>]), which we call the *compositum* M<sup>1</sup> •M<sup>0</sup> M<sup>2</sup> of p<sup>1</sup> : M<sup>1</sup> → M<sup>0</sup> and p<sup>2</sup> : M<sup>2</sup> → M0. Note that M<sup>00</sup> is a manifold since <sup>i</sup> acts properly discontinuously without fixed points on M-0. It is important to notice that the construction of M<sup>00</sup> and the covering maps M<sup>00</sup> → Mi, i = 1, 2, depend on the actual maps p1, p2, not just the spaces M0, M1, M2, but it is customary to leave out the maps from the notation if they are clear. If necessary, we will write M1 • <sup>p</sup>1 M0,p2 M2 (or M<sup>1</sup> •M0,p<sup>2</sup> M<sup>2</sup> if p<sup>1</sup> is clear).

### **2.3 Fiber Product**

Given a diagram (1.1), we can also form the (orbifold) *fiber product* M<sup>1</sup> ×M<sup>0</sup> M<sup>2</sup> (or, more precisely, M1 <sup>p</sup> ×1 M0,p2 M2 , where we use the same convention as before concerning including or leaving out the maps from the notation). We follow the construction of Thurston [ 95, 13.2.4] as explained in [25, 4.6.1] (but where we are using left actions instead of right actions): M-

$$M\_1 \times\_{M\_0} M\_2 = \Gamma\_0 \backslash \left( \dot{M}\_0 \times \Gamma\_1 \backslash \Gamma\_0 \times \Gamma\_2 \backslash \Gamma\_0 \right),$$

where the left action of γ<sup>0</sup> ∈ -<sup>0</sup> is given by

$$\gamma\_0(\mathbf{x}, \Gamma\_1 \boldsymbol{\gamma}\_0^{(1)}, \Gamma\_2 \boldsymbol{\gamma}\_0^{(2)}) = (\boldsymbol{\gamma}\_0(\mathbf{x}), \Gamma\_1 \boldsymbol{\gamma}\_0^{(1)} \boldsymbol{\gamma}\_0^{-1}, \Gamma\_2 \boldsymbol{\gamma}\_0^{(2)} \boldsymbol{\gamma}\_0^{-1})$$

for x ∈ M-<sup>0</sup>, γ (i) <sup>0</sup> ∈ -<sup>0</sup> (i = 1, 2). This has the universality property required for fiber products. Using the bijection

$$(\Gamma\_1 \backslash \Gamma\_0 \times \Gamma\_2 \backslash \Gamma\_0) \backslash \Gamma\_0 \to \Gamma\_1 \backslash \Gamma\_0 \backslash \Gamma\_2 \colon [(\alpha, \beta)] \to [\alpha \beta^{-1}],$$

we find an isometry

$$
\Gamma\_0 \times \Gamma\_2 \backslash \Gamma\_0 \backslash \Gamma\_0 \to \Gamma\_1 \backslash \Gamma\_0 / \Gamma\_2 \colon [(\alpha, \beta)] \to [\alpha]
$$

ry

$$
M\_1 \times\_{M\_0} M\_2 = \bigsqcup\_{\gamma \in \Gamma\_1 \backslash \Gamma\_0 / \Gamma\_2} (\Gamma\_1 \cap \gamma \Gamma\_2 \mathcal{Y}^{-1}) \backslash \widetilde{M}\_0.
$$

In our situation, where M<sup>1</sup> and M<sup>2</sup> are manifolds, it follows that M1×M0M<sup>2</sup> is also a (possibly disconnected) manifold, since elements of -<sup>1</sup> ∩γ -<sup>2</sup>γ <sup>−</sup><sup>1</sup> act without fixed points on M-0. The fiber product is a Riemannian manifold for the metric inherited from the universal covering (manifold) M-0.

The fiber product contains the compositum as a connected component. However, it is not necessarily connected (similar to the tensor product of two number fields not always being isomorphic to their compositum); in fact, we see from the above that M<sup>1</sup> ×M<sup>0</sup> M<sup>2</sup> has |-1\-0/ -<sup>2</sup>| connected components. The components need not all be isometric to the compositum, but if Mi → M<sup>0</sup> are Galois covers, then they are (since then, γ iγ <sup>−</sup><sup>1</sup> <sup>=</sup> <sup>i</sup> for all γ ∈ -0, i = 1, 2). M-

There are two projections

$$\begin{aligned} M\_1 \times\_{M\_0} M\_2 &\rightharpoonup \Gamma\_0 \backslash \left( \ddot{M}\_0 \times \Gamma\_i \backslash \Gamma\_0 \right) \widetilde{\equiv} M\_i, \end{aligned}$$

where the latter isometry is given by

$$\begin{aligned} &(\{\boldsymbol{\uprho} \boldsymbol{\uprho}(\boldsymbol{\upalpha}), \Gamma\_i \boldsymbol{\uprho}\_0^{(i)} \boldsymbol{\uprho}\_0^{-1}\}\_{\boldsymbol{\uprho} \in \Gamma\_0}) \mapsto \Gamma\_i \boldsymbol{\uprho}\_0^{(i)} \boldsymbol{\upalpha}, \\\\ &\mathbf{\uprho} \; \Gamma\_i \; \mathbf{has no fixed points on } \widetilde{M}\_0. \end{aligned}$$

which is bijective, since -0.

*Remark 2.3.1* There is a surjective map from the orbifold fiber product to the *settheoretic fiber product* 

$$M\_1 \times\_{M\_0} M\_2 \to \{ (\mathbf{x}\_1, \mathbf{x}\_2) \in M\_1 \times M\_2 \colon p\_1(\mathbf{x}\_1) = p\_2(\mathbf{x}\_2) \},\tag{2.2}$$

$$(\{\boldsymbol{\nu}\_0(\mathbf{x}), \boldsymbol{\Gamma}\_1 \boldsymbol{\nu}\_0^{(1)} \boldsymbol{\nu}\_0^{-1}, \boldsymbol{\Gamma}\_2 \boldsymbol{\nu}\_0^{(2)} \boldsymbol{\nu}\_0^{-1}\}\_{\boldsymbol{\nu}\_0 \in \boldsymbol{\Gamma}\_0}) \mapsto (\boldsymbol{\Gamma}\_1 \boldsymbol{\nu}\_0^{(1)} \boldsymbol{x}, \boldsymbol{\Gamma}\_2 \boldsymbol{\nu}\_0^{(2)} \boldsymbol{x}).$$

$$\text{on of } \boldsymbol{\Gamma}\_0 \text{ on } \widetilde{M}\_0 \text{ has fixed points, then this map is not necessarily injective.}$$

If the action of -<sup>0</sup> has fixed points, then this map is not necessarily injective, so the set-theoretic description of fiber product cannot be used in the orbifold setting, even if M1, M<sup>2</sup> are manifolds. However, if M<sup>0</sup> is itself a manifold (so -<sup>0</sup> acts without fixed points), then the map in (2.2) is an isometry of Riemannian manifolds (in particular, bijective), where the right hand side is a manifold since the projection maps are Riemannian submersions.

We can complete the diagram (1.1) into diagrams of Riemannian coverings

**Lemma 2.3.2** *If* Mi → M<sup>0</sup> *are Galois covers, then* M<sup>1</sup> ×M<sup>0</sup> M<sup>2</sup> *is connected and hence isometric to the compositum* M<sup>1</sup> •M<sup>0</sup> M<sup>2</sup> *if and only if* -0 = -1, -<sup>2</sup>*, the subgroup of* -<sup>0</sup> *generated by* -<sup>1</sup> *and* -<sup>2</sup>*. This holds if the degrees of* Mi → M<sup>0</sup> *are coprime.* 

*Proof* The number of components of the compositum is the cardinality of the double coset space -1\-0/ -2, and this is 1 if and only if -<sup>0</sup> = -1-2. Since <sup>i</sup> are normal subgroups of -0, the product -1-<sup>2</sup> is a subgroup; in fact, -1-2 = -1, -2. Hence the first statement holds. To see the second, note that we have a sequence of finite index group inclusions

$$
\Gamma\_l \hookrightarrow \langle \Gamma\_1, \Gamma\_2 \rangle \hookrightarrow \Gamma\_0,
$$

so we find that [-0 : -1, -<sup>2</sup>] divides gcd([-0, -1],[-0, -<sup>2</sup>]).

**Lemma 2.3.3** *If* Mi → M<sup>0</sup> *are* Gi*-Galois of coprime degree, then* 

(i) M<sup>1</sup> •M<sup>0</sup> M<sup>2</sup> → M<sup>0</sup> *is* (G<sup>1</sup> × G2)*-Galois;* 

(ii) M<sup>1</sup> •M<sup>0</sup> M<sup>2</sup> → M<sup>1</sup> *is* G2*-Galois and* M<sup>1</sup> •M<sup>0</sup> M<sup>2</sup> → M<sup>2</sup> *is* G1*-Galois.* 

*Proof* We always have an exact sequence

$$1 \to \Gamma\_1 \cap \Gamma\_2 \to \Gamma\_0 \xrightarrow{\varphi} \Gamma\_0 / \Gamma\_1 \times \Gamma\_0 / \Gamma\_2.$$

Now since [-<sup>0</sup> : -1∩-<sup>2</sup>] is divisible by the coprime integers [-<sup>0</sup> : -<sup>1</sup>] and [-<sup>0</sup> : -2], the map ϕ is surjective. This proves the first statement. The second statement follows from

$$
\Gamma\_1/(\Gamma\_1 \cap \Gamma\_2) \cong (\Gamma\_0/(\Gamma\_1 \cap \Gamma\_2))/(\Gamma\_0/\Gamma\_1) \cong \Gamma\_0/\Gamma\_2,
$$

and similarly with the indices 1 and 2 interchanged.

### **2.4 Normal Closure**

If M<sup>00</sup> - M<sup>0</sup> is a general finite covering of connected spaces, then there exists a connected space M and a sequence of coverings M - M<sup>00</sup> - M<sup>0</sup> such that M - M<sup>0</sup> and M - M<sup>00</sup> are finite and Galois (i.e., the corresponding subgroup of the fundamental group is normal), cf. [101, Thm. 1]. We call the minimal such M the *normal closure* of M<sup>00</sup> - M0. In terms of fundamental groups, if M<sup>00</sup> corresponds to the subgroup -<sup>00</sup> of -0, then M corresponds to the subgroup of -<sup>0</sup> given as the intersection of all -0-conjugates of -00, the so-called *(normal) core* of -<sup>00</sup> in -0. Alternatively, the normal core is the kernel of the action of -<sup>0</sup> by permutation on the cosets of -<sup>00</sup> in -0. In particular, since the index of -<sup>00</sup> in -<sup>0</sup> is finite, so is the index of the normal core, and M → M<sup>0</sup> is indeed finite. In fact, by the above

alternative description, the degree of M → M0, the index of the normal core in -0, is bounded by the order of the group of permutations of the set -00\-0. The number of elements of this set is the degree of the map M<sup>00</sup> - M0, and hence we find a bound

$$\{\Gamma\_0 : \Gamma\} \le \deg(M\_{00} \twoheadrightarrow M\_0) ! = \{\Gamma\_0 : \Gamma\_{00}\} !. \tag{2.3}$$

**Proposition 2.4.1** *Given manifolds* M<sup>1</sup> *and* M<sup>2</sup> *fitting into a diagram* (1.1)*, there exists a diagram of the form* (1.2)*.* 

*Proof* We start with a diagram (1.1) and add the normal closure of the compositum, to find a diagram of the form (1.2), where M corresponds to the normal core of -<sup>1</sup> ∩ -2: \M-:=

$$M := \Gamma \backslash \widetilde{M}\_0 \text{ for } \Gamma := \bigcap\_{\mathcal{Y} \in \Gamma\_0} \mathcal{Y}(\Gamma\_1 \cap \Gamma\_2) \mathcal{Y}^{-1} \dots$$

### **2.5 Commensurability**

Two manifolds M<sup>1</sup> and M<sup>2</sup> are called *commensurable* if they admit a common finite covering. Proposition 2.4.1 implies that if M<sup>1</sup> and M<sup>2</sup> have a common finite (developable orbifold) quotient as in diagram (1.1), then they are commensurable. We briefly look at the converse statement.

**Lemma 2.5.1** *Assume that* M<sup>1</sup> *and* M<sup>2</sup> *are commensurable with common finite covering* M*, let* <sup>M</sup> *denote the universal cover of* M *with isometry group I , and let* -, -1, -<sup>2</sup> *denote the subgroups of I corresponding to* M,M1, M<sup>2</sup> *respectively. Then a diagram of the form* (1.1) *exists if and only if* -<sup>1</sup> *and* -<sup>2</sup> *are of finite index in the subgroup of generated by* -<sup>1</sup> *and* -2*.*  <sup>0</sup> of *<sup>I</sup>* , then <sup>M</sup>-0 = M-

*Proof* If M<sup>0</sup> exists and corresponds to a subgroup and -1, -<sup>2</sup> is a finite index subgroup of -0, and by assumption -<sup>1</sup> and -<sup>2</sup> have finite index in -0. Conversely, if -1, -<sup>2</sup> is of finite index in -, we can set M<sup>0</sup> to be the corresponding orbifold -1, -2\M-. **Proposition 2.5.2** *If* M-

 *is a homogeneous space for a connected semisimple noncompact real Lie group with trivial center and no compact factors and* M<sup>1</sup> *and* M<sup>2</sup> *are quotients of* M *corresponding to commensurable irreducible uniform lattices* -1 *and* -2*, then there exists a diagram of the form* (1.1) *if at least one of* -<sup>1</sup> *and* -<sup>2</sup> *is non-arithmetic. In this case both* -<sup>1</sup> *and* -<sup>2</sup> *are non-arithmetic. Proof* Let *<sup>I</sup>* denote the isometry group of M-

. Since -<sup>1</sup> and -<sup>2</sup> are commensurable, their commensurator in *I* ,

$$\mathcal{H}^{\ell} := \mathbf{Comm}\_{\mathcal{J}}(\Gamma\_l) = \{ \mathbf{g} \in \mathcal{J} : \| \mathbf{g} \Gamma\_l \mathbf{g}^{-1} : \Gamma\_l \cap \mathbf{g} \Gamma\_l \mathbf{g}^{-1} \} \cdot \| \Gamma\_l : \Gamma\_l \cap \mathbf{g} \Gamma\_l \mathbf{g}^{-1} \| < \infty \}$$

is the same (indeed, if -<sup>1</sup> and -<sup>2</sup> are commensurable and -<sup>1</sup> is commensurable to g-<sup>1</sup>g−1, then also g-<sup>1</sup>g−<sup>1</sup> and g-<sup>2</sup>g−<sup>1</sup> are commensurable and, therefore -2 and g-<sup>2</sup>g−<sup>1</sup> are commensurable since commensurability is an equivalence relation). Margulis' theorem [66, Thm. (1), p. 2] (see also [102, Props. 6.2.4, 6.2.5 and Thm. 6.2.6]) states that either <sup>i</sup> is not arithmetic and of finite index in *C* or <sup>i</sup> is arithmetic and *C* is dense in *I* (to connect to the more general formulation in [66]: we consider a single semisimple group over the reals, and the "anisotropy condition" in loc. cit. is satisfied since we assume the group is not compact). This directly implies that either both lattices -<sup>1</sup> and -<sup>2</sup> are arithmetic or they are both not arithmetic.

Consider the sequence of inclusions

$$
\Gamma\_l \hookrightarrow \langle \Gamma\_1, \Gamma\_2 \rangle \hookrightarrow \emptyset \tag{2.4}
$$

for i = 1, 2.

If neither of <sup>i</sup> is arithmetic, the composed inclusion is of finite index. Hence the same holds for the first inclusion, and we can set M<sup>0</sup> := -1, -2\M-.

*Example 2.5.3* The proposition applies in particular to compact hyperbolic manifolds Mi = <sup>i</sup>\Hn, where <sup>H</sup><sup>n</sup> is hyperbolic n-space. ♦

**Proposition 2.5.4** *There exist commensurable compact hyperbolic Riemann manifolds* M<sup>1</sup> *and* M<sup>2</sup> *of dimensions* 2 *and* 3*, for which a diagram of the form* (1.1) *does not exist.* 

*Proof* The uniform arithmetic isospectral, non-isometric lattices constructed by Vignéras satisfy this property [97]; for commensurability, see [96, Ch. IV]. Additional information can be found in [24, Prop. 3].

*Remark 2.5.5* More generally, Alan Reid has shown that isospectral manifolds corresponding to arithmetic lattices are commensurable ([81], compare [61] for a quantitative statement). It is an open problem whether isospectral Riemann surfaces are always commensurable. Through work of Lubotzky, Samuels and Vishne, it is known that isospectrality does not imply commensurability [64] in general.

### **Open Problem**

Find further relations (other than Proposition 2.5.2) between commensurability and existence of a common finite quotient for special types of spaces, e.g., spaces whose universal covering is fixed and has infinite fundamental group. Example 2.5.3 concerns the case where the common universal covering is hyperbolic n-space.

### **Open Problem**

Find a criterion to decide precisely which pairs of arithmetic manifolds are common finite cover of some developable orbifold. See Remark 2.1.1 for a relation between this problem for surfaces and branching data.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 3 Spectra, Group Representations and Twisted Laplacians**

*We're gonna do the twist and it goes like this*  — "Let's twist again", written by Kal Mann and Dave Appell

In this chapter, we review basic notions about spectra, group representations, and twisted Laplace operators. We first recall how to define the spectrum and the spectral zeta function for a general symmetric second order elliptic differential operator acting on smooth sections of a Hermitian line bundle. We prove that the non-zero spectrum (i.e., the spectral zeta function) determines the entire spectrum on an odddimensional manifold, but also give an example showing that this is not always true for even-dimensional manifolds; the example is obstructed by the non-vanishing of some topological genus. After setting up some notation from representation theory, we discuss *G*-sets and weak conjugacy ("Gaßmann equivalence") of subgroups of a group, explaining the interrelations. In the final sections, we introduce twisted Laplacians, corresponding to unitary representations of the fundamental group. After this, we focus on the case of a twisted Laplacian arising from a finite Galois cover of manifolds and we relate the spectrum on the top manifold to that of the induced representation on the bottom manifold. We relate the multiplicity of zero in the spectrum to the multiplicity of the trivial representation in the given representation, and finally we show that, contrary to the general case, the multiplicity of zero in the spectrum of a twisted Laplacian is determined from the non-zero spectrum, provided one also knows the usual Laplace spectrum of the manifold.

As basic background references for this chapter, we use [54, 83, 88].

### **3.1 Spectrum and Spectral Zeta Function**

Let *M* = *(M, g)* denote a connected closed oriented smooth Riemannian manifold with Riemannian metric *g*. Let *E* denote a Hermitian bundle on *M* and *A* a symmetric second order elliptic differential operator acting on smooth sections *C*∞*(M, E)* of *E* with non-negative eigenvalues. The operator extends to the corresponding space *L*2*(M, E)* of *L*2-sections where it has a dense domain. The *spectrum σM(A) of A* (or *σ (A)* is *M* is fixed) is the multiset of eigenvalues of *A*, where the multiplicities of the elements in the set are given by the multiplicities of the eigenvalues.

We make the following convenient notational conventions: if *S*<sup>1</sup> and *S*<sup>2</sup> are multisets, we let *S*<sup>1</sup> ∪*S*<sup>2</sup> denote the multiset consisting of elements of *S*<sup>1</sup> or *S*2, where the multiplicity of an element is the sum of the multiplicities of that element in *S*<sup>1</sup> and *S*2, and for a multiset *S* and an integer *n*, we mean by *nS* the multiset of elements of *S* where all multiplicities are multiplied by *n*.

*Example 3.1.1* To *(M, g)* is associated a *Laplace(–Beltrami) operator M*, acting on the space *C*∞*(M)* of smooth functions on *M*, given in local coordinates *(x<sup>i</sup> )* as

$$\Delta\_M(f) := -\det(g)^{-1/2} \sum\_{i,j} \frac{d}{d\boldsymbol{\alpha}^j} \left( \det(g)^{1/2} g^{ij} \frac{d}{d\boldsymbol{\alpha}^j} f \right) .$$

An intrinsic definition follows from the more general next example. ♦

*Example 3.1.2* More generally, for every *k* ≥ 0, there is such a *Laplace operator k <sup>M</sup> acting on (the space of) k-forms C*∞*(M, <sup>k</sup> <sup>T</sup>* <sup>∗</sup>*M)*, defined as follows. If *<sup>d</sup><sup>k</sup>* : *<sup>C</sup>*∞*(M, <sup>k</sup> <sup>T</sup>* <sup>∗</sup>*M)* <sup>→</sup> *<sup>C</sup>*∞*(M, k*+<sup>1</sup> *<sup>T</sup>* <sup>∗</sup>*M)* denotes the exterior derivative on *<sup>k</sup>*-forms, and *δk*+<sup>1</sup> : *<sup>C</sup>*∞*(M, k*+<sup>1</sup> *<sup>T</sup>* <sup>∗</sup>*M)* <sup>→</sup> *<sup>C</sup>*∞*(M, <sup>k</sup> <sup>T</sup>* <sup>∗</sup>*M)* the adjoint of *d<sup>k</sup>* for the inner product induced by the metric *g*, then *<sup>k</sup> <sup>M</sup>* <sup>=</sup> *<sup>δ</sup>k*+1*d<sup>k</sup>* <sup>+</sup> *<sup>d</sup>k*−1*δ<sup>k</sup>* acting on *C*∞*(M, <sup>k</sup> <sup>T</sup>* <sup>∗</sup>*M)* with *d*−<sup>1</sup> <sup>=</sup> <sup>0</sup> by convention.

For *<sup>k</sup>* <sup>=</sup> 0, *<sup>k</sup> <sup>M</sup>* equals <sup>0</sup> *<sup>M</sup>* <sup>=</sup> *<sup>δ</sup>*1*<sup>d</sup>* <sup>=</sup> *M* as defined in Example 3.1.1. The kernel of *<sup>k</sup> <sup>M</sup>* consists of so-called *harmonic k-forms*, which, by a theorem of Hodge, is identified with H*<sup>k</sup>* dR*(M)*, the *k*-th de Rham cohomology group of *M*. De Rham's theorem identifies H*<sup>k</sup>* dR*(M)* ∼= <sup>H</sup>*k(M,* **<sup>R</sup>***)* with the usual real-valued (singular) cohomology of *M*. Thus, the multiplicity of 0 in the spectrum of *<sup>k</sup>* equals the *k*-th Betti number of *M*:

$$\text{mult}\_0(\sigma\_M(\Delta\_M^k)) = \dim\_\mathbb{R} \mathcal{H}^k(M, \mathbf{R}).\tag{3.1}$$

We refer to [83, Chapter 1] for details. ♦

The *spectral zeta function* of *A* as above is defined as

$$\zeta\_{M,A}(s) = \zeta\_A(s) := \sum\_{0 \neq \lambda \in \sigma(A)} \lambda^{-s}$$

with the sum not involving the zero eigenvalues. The function can be meromorphically continued to the entire complex plane [83, Thm. 5.2]. Since *ζA(s)* is a (generalised) Dirichlet series, the identity theorem for such Dirichlet series [46,

Thm. 6] implies that it is determined by its values on a countable set with an accumulation point, e.g., by its values at all sufficiently large integers.

### **3.2 Spectrum Versus Spectral Zeta Function**

We will formulate all results using the spectrum, rather than the spectral zeta function. For odd-dimensional manifolds, these give exactly the same information, as the following proposition shows.

**Proposition 3.2.1** *If M is an odd-dimensional manifold, the multiset σM(A) and the function ζM,A(s) mutually determine each other.* 

*Proof* It is clear that the function *ζA(s)* determines *σ (A)* − {0}, so we only need to show that if *M* is of odd dimension, the multiplicity of zero in the spectrum is also determined by *ζA*; this multiplicity is dim ker *A*, which equals −*ζA(*0*)* if *M* has odd dimension (see [83, Thm. 5.2]). 

The result in Proposition 3.2.1 does not hold in general if *M* is of even dimension *n*, as the example in the following proposition shows.

**Proposition 3.2.2** *There exists a* 4*-dimensional manifold M* (*in fact, M may be chosen as a complex quartic surface*) *and two second order bundle operators* ± *on M such that ζ*<sup>+</sup> = *ζ*<sup>−</sup> *but σ (*+*)* = *σ (*−*).*

*Proof* The basic idea is that two commuting operators have the same non-zero spectrum, but that the difference of the dimensions of their kernels can have a topological interpretation as an index, that might be non-vanishing. The proof will use the theory of spin manifolds and the index theorem for the Dirac operator, for which we refer to [11] or [39].

Suppose *M* is an even dimensional spin manifold with Dirac operator

$$D = \begin{pmatrix} 0 & D^+ \\ D^- & 0 \end{pmatrix}.$$

where the spinor bundle is decomposed into eigenspaces for the chirality operator as *S*<sup>+</sup> ⊕ *S*−, with *D*<sup>+</sup> : *S*<sup>+</sup> → *S*<sup>−</sup> and *D*<sup>−</sup> : *S*<sup>−</sup> → *S*+*, D*<sup>−</sup> = *(D*+*)*<sup>∗</sup> adjoint to *D*+. Then the second order operators

$$
\Delta^{\pm} := D^{\mp}D^{\pm}
$$

have the same non-zero spectrum (this is true in general for the non-zero spectrum of the products *AB* and *BA* of two operators *A* and *B*); hence *ζ*<sup>+</sup> = *ζ*<sup>−</sup> .

On the other hand,

$$\ker \Delta^{+} = \ker (D^{+})^{\*}D^{+} = \ker D^{+} $$

since if +*ϕ* = *(D*+*)*∗*D*+*ϕ* = 0, then

$$||D^+\varphi||^2 = \langle D^+\varphi, D^+\varphi\rangle = \langle \varphi, (D^+)^\*D^+\varphi\rangle = 0.$$

Hence with *m*<sup>±</sup> := mult0*(σ (*±*))* the multiplicity of 0 in the spectrum of ±, we find that

$$m^{+} - m^{-} = \dim \ker \Delta^{+} - \dim \ker \Delta^{-} = \dim \ker D^{+} - \dim \ker D^{-}$$

$$= \text{index } D = \int\_{M} \widehat{A}(M)$$

is, by the Atiyah–Singer index theorem, the *A* -genus of *M*, which may be non-zero (compare [11, 3.4, 4.1]). For example, if *M* is a complex quartic surface (of real dimension 4) in **CP**3, then index *<sup>D</sup>* <sup>=</sup> *A(M)* <sup>=</sup> <sup>2</sup> (see, e.g., [39, p. 727]). This shows that *σ (*+*)* = *σ (*−*).* 

*Remark 3.2.3* If one is willing to consider orbifolds instead of manifolds, there exist two-dimensional orbifolds that are isospectral for the Laplace operator acting on 1-forms (cf. Example 3.1.2), as shown by the following example of Gordon and Rossetti. Consider the quotient of the standard flat torus **Z**2\**R**<sup>2</sup> by the involutions induced by the following maps on **R**<sup>2</sup> (with coordinates *(x, y)*):


Then the non-zero spectra of the Laplace operators acting on the space of 1-forms on *C,M* and O agree. However, 0 is not an eigenvalue on 1-forms for *C* and *M*, whereas it is for O [42, Example 2.5 and Theorem 3.1].

Interestingly, our main results, such as Theorem 1.2.1 and Proposition 1.2.5, are formulated in their strongest possible form using *precisely* the multiplicity of zero in the spectrum of certain twisted Laplacians. For these operators, it turns out that also in even dimension this multiplicity (and hence the zeta function) is fixed by the non-zero spectrum of the usual and the twisted Laplacian, cf. Proposition 3.10.1 below.

### **3.3 Group Representations**

If *<sup>G</sup>* is a finite group, let *<sup>G</sup>*<sup>q</sup> <sup>=</sup> Hom*(G,* **<sup>C</sup>**∗*)* denote the group of linear characters of *G*, and let Irr*(G)* denote the set of inequivalent irreducible unitary representations of *G*. We consider complex representations as **C**[*G*]-modules M or group homomorphisms *<sup>ρ</sup>* : *<sup>G</sup>* <sup>→</sup> Aut*(V )* ∼= GL*(N ,* **<sup>C</sup>***)* with *<sup>V</sup>* <sup>=</sup> **<sup>C</sup>***<sup>N</sup>* and freely mix these concepts, writing expressions such as "M ∼= *ρ*". By further slight abuse of notation, if *ρ*<sup>1</sup> and

*ρ*<sup>2</sup> are representations of *G*, we write

$$\langle \rho\_1, \rho\_2 \rangle = \langle \text{tr}(\rho\_1(-)), \text{tr}(\rho\_2(-)) \rangle = \frac{1}{|G|} \sum\_{\mathcal{g} \in G} \text{tr}(\rho\_1(\mathcal{g})) \overline{\text{tr}(\rho\_2(\mathcal{g}))}$$

for the inner product of the corresponding characters in the space of class functions. The multiplicity of an irreducible representation *ρ* ∈ Irr*(G)* in the decomposition into irreducibles of a general representation *ρ* of *G* is then *ρ,ρ* .

The *regular representation ρG,*reg corresponds to the **C**[*G*]-module **C**[*G*]. It decomposes as

$$\rho\_{G, \text{reg}} = \bigoplus\_{\rho\_l \in \text{Irr}(G)} \dim(\rho\_l)\rho\_l.$$

If *H* is a subgroup of *G* and *ρ* a representation of *H*, then Ind*<sup>G</sup> <sup>H</sup> ρ* denotes the representation induced by *ρ* from *H* to *G*: if *ρ* corresponds to the **C**[*H*]-module *V* , then Ind*<sup>G</sup> <sup>H</sup> ρ* corresponds to the **C**[*G*] module *W* := **C**[*G*] ⊗**C**[*H*] *V* . In coordinates, this means the following: since *G* permutes the cosets of *H* in *G*, if we choose coset representatives

$$G/H = \{ \mathbf{g}\_1 H = H, \dots, \mathbf{g}\_n H \},$$

then for any *g* ∈ *G* we have

$$\mathbf{g} \mathbf{g}\_{l} = \mathbf{g}\_{\mathbf{g}(l)} h\_{\mathbf{g},l}$$

for some *hg,i* ∈ *H* and some permutation *i* → *g(i)* of {1*,...,n*}. If we write

$$W = V^{G/H} = \bigoplus\_{i=1}^{n} \mathcal{g}\_i V$$

using *gi* as placeholder, then with *vi* ∈ *V* , we have

$$\operatorname{Ind}\_{H}^{G}\rho(\mathbf{g})\left(\sum\_{l}\mathbf{g}\_{l}v\_{l}\right)=\sum\_{l}\mathbf{g}\_{\mathcal{S}(l)}\rho(h\_{\mathbf{g},l})(v\_{l}).\tag{3.2}$$

Let Res*<sup>G</sup> <sup>H</sup> ρ* denotes the restriction of *ρ*, a representation of *G*, from *G* to *H*. If *ρ* is a representation, *ρ* denotes the complex conjugate representation. Recall the standard calculation rules

$$<\langle \rho \bowtie \rho \rhd \rho \rhd \langle \rho \rhd \rho \rhd \overline{\rho \rhd} \rangle \text{ and } \langle \text{Ind}\_H^G \rho \rhd , \rho \rhd \rangle = \langle \rho \rhd \text{Res}\_H^G \rho \rhd \rho \rhd \rho$$

(the latter is known as "Frobenius reciprocity").

More generally, the above theory applies *mutatis mutandis*, replacing **C** by an algebraically closed field of characteristic coprime to the order |*G*| of the group *G*. For a non-algebraically closed field *K* of characteristic coprime to |*G*|, an irreducible representation might decompose over the algebraic closure into a sum of irreducible (Galois-conjugate) representations, but the above theory remains valid, with the caveat that a rational character is not always the character of a rational representation, but a multiple is. If the characteristic of *K* divides |*G*|, the category of *K*[*G*]-modules is not semisimple, so complementary modules for submodules do not always exist. For us, it will be important that, in general, the regular representation is defined over **Q**, and induction and restriction turn *K*representations into *K*-representations.

### **3.4** *G***-Sets**

If *G* is a group, a *G-set* is a set that admits a left *G*-action. An example is the left cosets *G/H* of a subgroup *H* with the action of left multiplication by *G*.A *morphism* of *G*-sets is a *G*-equivariant map of the sets. We say a *G*-set is *transitive*  if *G* acts transitively on it. If *X* is a transitive *G*-set and *H* the stabiliser of any point in *X*, then *X* is isomorphism to *G/H* as *G*-set.

### **3.5 (Weak) Conjugacy**

We let **1** = **1***<sup>G</sup>* denote the trivial representation of a group *G*. If, as before, {*g*1*,...,gn*} is a set of representatives for the (left) *H*-cosets in *G*, then Ind*<sup>G</sup> <sup>H</sup>* **1** is the permutation representation (i.e., the action of each *g* ∈ *G* is given by a permutation matrix, a matrix having exactly one non-zero entry 1 in each row and column) given by the action of *G* on the vector space

$$\mathbf{C}[G/H] := \bigoplus\_{l=1}^{n} \mathbf{C}\_{\mathcal{G}\_l} H$$

spanned by the cosets of *H* in *G*. We have the following.

**Proposition 3.5.1** *Let H*<sup>1</sup> *and H*<sup>2</sup> *denote two subgroups of a finite group G.* 

	- (a) *The representations* Ind*<sup>G</sup> <sup>H</sup>*<sup>1</sup> **<sup>1</sup>** ∼= Ind*<sup>G</sup> <sup>H</sup>*<sup>2</sup> **1** *are isomorphic.*
	- (b) *Each conjugacy class c of G intersects H*<sup>1</sup> *and H*<sup>2</sup> *in the same number of elements.*
	- (c) *There exists a set-theoretic bijection ψ* : *H*<sup>1</sup> → *H*<sup>2</sup> *such that h*<sup>1</sup> *and ψ(h*1*) are conjugate in G for any h*<sup>1</sup> ∈ *H*1*.*

*If any of these holds, we say H*<sup>1</sup> *and H*<sup>2</sup> *are* weakly conjugate *in G.* 

(ii) *The stronger property that the groups H*<sup>1</sup> *and H*<sup>2</sup> *are* conjugate *in G is equivalent to the cosets G/H*<sup>1</sup> *and G/H*<sup>2</sup> *being isomorphic as G-sets.* 

Weak conjugacy is sometimes called "almost conjugacy", and also known as "Gaßmann equivalence" in number theory, cf. [77].

### *Proof*

(i) Representation isomorphism is the same as isomorphism of characters, and the character of the representation Ind*<sup>G</sup> <sup>H</sup>* **1** is

$$\psi(\mathbf{g}) = |\{\mathbf{g}\} \cap H| \cdot |C\_G(\mathbf{g})|/|H|,\tag{3.3}$$

where [*g*] is the conjugacy class of *g* and *CG(g)* is the centraliser of *g* in *G* (compare, e.g., [18, §1]). The final equivalent statement is proven in [24, Lemma 2].

(ii) The existence of a *G*-isomorphism *φ* : *G/H*<sup>1</sup> → *G/H*<sup>2</sup> implies that the *G*stabiliser *H*<sup>1</sup> of the coset *eH*<sup>1</sup> equals the *G*-stabiliser *gH*2*g*−<sup>1</sup> of some coset *gH*<sup>2</sup> <sup>=</sup> *φ(eH*1*)*, and hence *H*<sup>1</sup> <sup>=</sup> *gH*2*g*−1. Conversely, if *H*<sup>2</sup> <sup>=</sup> *<sup>g</sup>*−<sup>1</sup> <sup>0</sup> *H*1*g*0, the map *φ* : *G/H*<sup>1</sup> → *G/H*2, *φ(gH*1*)* = *gg*0*H*<sup>2</sup> is a well-defined isomorphism of *G*-sets. 

*Remark 3.5.2* If we have a diagram (1.1) and *M*<sup>1</sup> and *M*<sup>2</sup> have the same Laplace spectrum (viz., the same spectral zeta function) and the same dimension *n*, then they have the same volume (from Weyl's law, or, equivalently, from the value of the residue of their zeta functions at *s* = *n/*2). Since the covering degrees deg*(pi)* of *pi* are vol*(Mi)/*vol*(M*0*)*, these are also equal, and hence in diagram (1.2), we find from |*Hi*|=|*G*|*/* deg*(pi)* that |*H*1|=|*H*2|. So in this case, there is *always* a set-theoretic bijection *ψ* : *H*<sup>1</sup> → *H*2.

### **3.6 Twisted Laplacian**

Suppose *ρ* : *π*1*(M)* → U*(N ,* **C***)* is a unitary representation of the fundamental group *π*1*(M)* of *M*. Let : *<sup>M</sup>* <sup>→</sup> *<sup>M</sup>* denote the universal covering of *M*, and set *Eρ* := *<sup>M</sup>* <sup>×</sup>*<sup>ρ</sup>* **<sup>C</sup>***<sup>N</sup>* , where the subscript *<sup>ρ</sup>* indicates equivalence classes for the relation

$$(z, v) \sim (\wp z, \rho(\wp)^{-1} v)^{\flat}$$

for any *<sup>z</sup>* <sup>∈</sup> *M,v* <sup>∈</sup> **<sup>C</sup>***<sup>N</sup>* and *<sup>γ</sup>* <sup>∈</sup> *<sup>π</sup>*1*(M)*. Now *Eρ* is a flat vector bundle of rank *<sup>N</sup>* over *M*, whose global sections *f* ∈ *C*∞*(M, Eρ)* correspond bijectively to smooth *<sup>ρ</sup>*-equivariant vector-valued functions #–*<sup>f</sup>* : *<sup>M</sup>* <sup>→</sup> **<sup>C</sup>***<sup>N</sup>* , i.e., functions with #–*f (γ z)* <sup>=</sup> *ρ(γ )*#–*f (z)*. The *twisted Laplacian* 

$$
\Delta\_{M,\rho} = \Delta\_{\rho} \colon C^{\infty}(M, E\_{\rho}) \to C^{\infty}(M, E\_{\rho}),
$$

is defined as

$$
\overrightarrow{\Delta\_{\rho}(f)} := \Delta\_{\widetilde{M}} \overrightarrow{f}(z).
$$

Henceforth, we will also denote the spectrum *σM(ρ)* simply by *σM(ρ)* or by *σ (ρ)* if the underlying manifold *M* is fixed.

Notice that when *ρ* = *ρ*<sup>1</sup> ⊕*ρ*2, then *ρ* admits a block decomposition *ρ*<sup>1</sup> ⊕*ρ*<sup>2</sup> on *Eρ* ∼= *Eρ*<sup>1</sup> ⊕ *Eρ*<sup>2</sup> , and thus, the spectrum satisfies (as multisets)

$$
\sigma\_M(\rho\_1 \oplus \rho\_2) = \sigma\_M(\rho\_1) \cup \sigma\_M(\rho\_2).
$$

### **3.7 Twisted Laplacians on Finite Covers**

In case *ρ* factors through a finite group *G*, there is no need to use the universal covering. Let *M* → *M* denote a (fixed-point free) *G*-cover and *ρ* : *G* → U*(N ,* **C***)* a unitary representation. The vector space *C*∞*(M, Eρ)* is canonically isomorphic to the vector space of smooth *ρ*-equivariant vector-valued functions on *M* given by

$$\mathcal{C}^{\infty}\_{\rho}(M', \mathbb{C}^{N}) := \{ \overrightarrow{f} \in \mathcal{C}^{\infty}(M', \mathbb{C}^{N}) \mid \overrightarrow{f}(\operatorname{\boldsymbol{\chi}} \mathbf{x}) = \rho(\operatorname{\boldsymbol{\chi}}) \overrightarrow{f}(\mathbf{x}), \forall \mathbf{x} \in M', \boldsymbol{\chi} \in G \}.$$

In this case,

$$
\overrightarrow{\Delta\_{\rho}f} = \Delta\_{M'}\overrightarrow{f}\,,\tag{3.4}
$$

where #–*<sup>f</sup>* is the *ρ*-equivariant function in *C*<sup>∞</sup> *<sup>ρ</sup> (M ,* **C***<sup>N</sup> )* corresponding to *f* . Note that *M* #–*<sup>f</sup>* is again a *ρ*-equivariant function in *C*<sup>∞</sup> *<sup>ρ</sup> (M ,* **C***<sup>N</sup> )* and therefore represents an element in *C*∞*(M, Eρ)*.

### **3.8 Twisted Laplacians for Induced Representations**

The following lemma is stated in [91, Lemma 1]; we write a proof using our notation.

**Lemma 3.8.1** *If M* → *M*<sup>1</sup> → *M*<sup>0</sup> *is a tower of finite Riemannian coverings and M* → *M*<sup>0</sup> *is Galois with group G, M* → *M*<sup>1</sup> *with group H, and ρ* : *H* → U*(N ,* **C***) a representation, then* 

$$
\sigma\_{M\_0}(\mathrm{Ind}\_H^G \rho) = \sigma\_{M\_1}(\rho).
$$

*Proof* Write *ρ*<sup>∗</sup> := Ind*<sup>G</sup> <sup>H</sup> ρ* : *G* → U*(Nn,* **C***)* and let

$$G/H = \{ \mathbf{g}\_1 H = H, \dots, \mathbf{g}\_n H \}$$

denote representatives for the distinct cosets of *H* in *G*. Define two maps

by

$$\Phi(\overrightarrow{f})(\mathbf{x}) = (\overrightarrow{f}(\mathbf{g}\_1^{-1}(\mathbf{x})), \dots, \overrightarrow{f}(\mathbf{g}\_n^{-1}(\mathbf{x})))$$

and

$$\Psi(\overrightarrow{F}) = \Psi((\overrightarrow{f\_1}, \dots, \overrightarrow{f\_n})) := \overrightarrow{f\_1}.$$

Recall that by definition

$$(\rho^\*(\mathbf{g})((\overrightarrow{f\_l}(\mathbf{x}))\_{l=1}^n) = (\rho(\mathbf{g}\_{\mathcal{g}(l)}^{-1}\mathbf{g}\mathbf{g}\_l)\overrightarrow{f}\_{\mathcal{g}(l)}(\mathbf{x}))\_{l=1}^n$$

where *g(i)* is given by

$$\operatorname{gg}\_{\ell} H = \operatorname{g}\_{\operatorname{g}(\ell)} H.$$

This allows one to check that and are well defined and mutually inverse bijections. Recall that # – *ρ(f )* <sup>=</sup> *M(* #–*f )* and # – *ρ*<sup>∗</sup> *(F )* <sup>=</sup> *M(* #–*F )* with *M* applied componentwise. Since the *gi* are isometries, is a unitary operator in *L*<sup>2</sup> and *M* = *M*, so that we have the intertwining

$$
\Phi \circ \Delta\_{\rho} = \Delta\_{\rho^\*} \circ \Phi,
$$

and the equality of spectra follows.

As is shown in [91], the main theorem of Sunada [90] follows easily from this.

**Theorem 3.8.2 (Sunada's Theorem [90, Theorem 1])** *If we have a diagram of the form* (1.2) *and H*<sup>1</sup> *and H*<sup>2</sup> *are weakly conjugate in G, then M*<sup>1</sup> *and M*<sup>2</sup> *are isospectral.*

*Proof* We apply Lemma 3.8.1 to the trivial representation *ρ* = **1** for *M* → *Mi* → *<sup>M</sup>*<sup>0</sup> with *<sup>i</sup>* <sup>=</sup> <sup>1</sup> and *<sup>i</sup>* <sup>=</sup> 2. Since *<sup>H</sup>*<sup>1</sup> and *<sup>H</sup>*<sup>2</sup> are weakly conjugate, Ind*<sup>G</sup> <sup>H</sup>*<sup>1</sup> **<sup>1</sup>** ∼= Ind*<sup>G</sup> <sup>H</sup>*<sup>2</sup> **1** (cf. Proposition 3.5.1), and so

$$
\sigma\_{M\_1} = \sigma\_{M\_1}(\mathbf{1}) = \sigma\_{M\_0}(\mathrm{Ind}\_{H\_1}^G \mathbf{1}) = \sigma\_{M\_0}(\mathrm{Ind}\_{H\_2}^G \mathbf{1}) = \sigma\_{M\_2}(\mathbf{1}) = \sigma\_{M\_2},
$$

finishing the proof.

*Remark 3.8.3* The original proof used a trace formula [90, Lemma 1], that is now hidden in the computations with induced representations and their characters. The trace formula proof has the advantage to apply to all "natural" operators alike, such as the Laplace operators on *k*-forms (cf. Example 3.1.2); see Sect. 4.2 for more on natural operators and strong isospectrality. For another exposition in the style of the original argument and background information, see [23, Chapter 11].

*Remark 3.8.4* The converse of the theorem is not true: isospectral manifolds fitting into a diagram of the form (1.2) do not necessarily have *H*<sup>1</sup> and *H*<sup>2</sup> weakly conjugate in *G*. See Corollary 4.2.5 for a spectral characterisation of weak conjugacy using twisted Laplacians.

*Remark 3.8.5* The main application of Theorem 3.8.2 is to the construction of isospectral, non-isometric manifolds. For this, one needs to realise a diagram of manifolds as in (1.2) *and* guarantee that *M*<sup>1</sup> and *M*<sup>2</sup> are not isometric, for example, by making sure that H1*(M*1*)* and H1*(M*2*)* are distinct.

### **3.9 Multiplicity of Zero in Twisted Laplace Spectra**

Decomposing a general representation *ρ* : *G* → U*(N ,* **C***)* into irreducibles as *ρ* = *ρi, ρρi*, we have

$$\sigma\_M(\Delta\_\rho) = \bigcup \langle \rho\_l, \rho \rangle \sigma\_M(\Delta\_{\rho\_l}).\tag{3.5}$$

Applied to the regular representation, we find a relation between the spectra of the usual Laplacian on the cover *M* and of the twisted Laplacians on the original manifold *M*, as follows:

$$
\sigma\_{M'} (\Delta\_{M'}) = \sigma\_M (\Delta\_{\rho\_{G, \text{reg}}}) = \bigcup \dim(\rho\_l) \sigma\_M (\Delta\_{\rho\_l}).\tag{3.6}
$$

(The first equality follows from Lemma 3.8.1 since Ind*<sup>G</sup>* {1} **<sup>1</sup>** <sup>=</sup> **<sup>C</sup>**[*G*] = *ρG,*reg). We see in particular that the eigenvalues of any twisted Laplacian are also eigenvalues of the usual Laplace operator of the corresponding cover.

**Lemma 3.9.1** *Let G be a finite group acting by fixed-point free isometries on a closed connected Riemannian manifold M with quotient M* = *G*\*M . If ρ is any unitary representation of G, then the multiplicity ρ ,* **1** *of the trivial representation*

*in the decomposition of ρ into irreducibles equals* dim ker *ρ, the multiplicity of the zero eigenvalue in σM(ρ).* 

*Proof (First Proof of Lemma 3.9.1)* Since *M* and *M* are connected, the multiplicity of zero in *σM(M)* and *σM(M)* is one. It follows from the decomposition of multisets (3.6) that for any irreducible representation *ρi* = **1** of *G*, *σM(ρi)* does not contain zero. If we now decompose *ρ* as a sum of irreducibles, the decomposition of multisets (3.5) implies that the multiplicity of zero in *σM(ρ)* is indeed the multiplicity with which **1** occurs in *ρ*. 

We can also give a "direct" proof, as follows.

*Proof (Second Proof of Lemma 3.9.1)* Let *ρ* : *G* → U*(N ,* **C***)*. A function

$$f \in \ker \Delta\_{\rho} \subseteq C^{\infty}(M, E\_{\rho})$$

corresponds to a function #–*<sup>f</sup>* on *<sup>M</sup>* with *M* #–*<sup>f</sup>* <sup>=</sup> <sup>0</sup> and #–*f (γ z)* <sup>=</sup> *ρ(γ )*#–*f (z)* for all *<sup>γ</sup>* <sup>∈</sup> . Since *M* is closed and connected, this implies that #–*<sup>f</sup>* <sup>=</sup> #–*<sup>f</sup>* <sup>0</sup> is a constant vector in **C***<sup>N</sup>* , and the equivariance condition translates into

$$(\rho(\gamma) - 1)\vec{\tilde{f}}\_0 = 0 \quad \text{for all } \mathcal{Y} \in G.$$

Hence each such linearly independent vector #–*f*<sup>0</sup> <sup>∈</sup> **<sup>C</sup>***<sup>N</sup>* can be used to split off a one-dimensional invariant subspace in *ρ*, and we find the result. 

### **3.10 Spectrum Versus Spectral Zeta Function for Twisted Laplacians**

In Proposition 3.2.2 we showed that, in general, on an even-dimensional manifold, knowledge of the spectrum is stronger than that of the spectral zeta function, i.e., of the non-zero spectrum. The operators in Proposition 3.2.2 were not twisted Laplacians. For twisted Laplace operators, the situation is better, as the following proposition shows.

**Proposition 3.10.1** *Let G be a finite group acting by fixed-point free isometries on a closed connected n-dimensional Riemannian manifold M with quotient M* = *G*\*M . If ρ is any unitary representation of G, then on the one hand the pair of multisets σM(ρ) and σM(), and on the other hand the pair of zeta functions ζM,ρ (s) and ζM,(s) mutually determine each other. In fact, the multiplicity of zero in σ (ρ) is given by* 

$$\dim \ker \Delta\_{\rho} = \left( \xi\_{\Delta}(0) + 1 \right) \left. \frac{\xi\_{\Delta\_{\rho}}(s)}{\xi\_{\Delta}(s)} \right|\_{s=\frac{\mu}{2}} - \xi\_{\Delta\_{\rho}}(0). \tag{3.7}$$

*Remark 3.10.2* As an illustration consider the easy situation when |*G*| = 1, so *<sup>M</sup>* <sup>=</sup> *<sup>M</sup>* and *<sup>ρ</sup>* <sup>∼</sup> *<sup>N</sup>***<sup>1</sup>** for some *N*. Then *σM(ρ)* <sup>=</sup> *NσM(<sup>N</sup> )* <sup>=</sup> *NσM*, *ζρ* <sup>=</sup> *Nζ* and mult0*(σ (N ))* <sup>=</sup> *<sup>N</sup>* is equal to *(ζ(*0*)* <sup>+</sup> <sup>1</sup>*) ζρ (s) ζ(s) <sup>s</sup>*<sup>=</sup> *<sup>n</sup>* 2 − *ζρ (*0*)* = 2*N* − *N.*

*Proof* It suffices to prove that *ζ* and *ζρ* determine the multiplicity of zero in the spectrum (i.e, the dimension of the kernel) of *ρ*. If the dimension *n* of *M* is odd, this follows from the stronger Proposition 3.2.1. For even *n* and a general operator *A* as in Sect. 3.1,

$$\zeta\_A(0) = -\dim \ker A + (4\pi)^{-n/2} \int\_M u\_{n/2}(A),\tag{3.8}$$

where tr*(e*−*tA)* <sup>∼</sup> *(*4*πt)*−*n/*<sup>2</sup> <sup>∞</sup> *k*=0 *<sup>M</sup> uk(A)t <sup>k</sup>* is the asymptotic expansion of the heat kernel of *A* as *t*↓0 [83, Thm. 5.2].

We apply this in our situation, with

$$A = \Delta\_{\rho} = \bigoplus\_{i=1}^{N} \Delta\_{M'}$$

acting on *C*∞*(M, Eρ)* = *C*<sup>∞</sup> *<sup>ρ</sup> (M ,* **C***<sup>N</sup> )*. Denote by 1*<sup>N</sup>* the identity matrix of size *N* × *N*. Recall that the principal symbol *p(M)* of a Laplace operator *M* on a Riemannian manifold *(M, g)* is determined by the metric tensor *g* (more accurately, it is the quadratic form on the cotangent bundle dual to *g*). Since *M* → *M* is a Riemannian covering, the metric tensor of *M* pulls back to that of *M* , and hence the principal symbol of *M* is the same as that of *M*. Therefore, *ρ* is a "Laplacestyle" operator in the sense of [40, 1.2]: it has (matrix) principal symbol the diagonal matrix *p(ρ)* = *p(M)* · 1*<sup>N</sup>* . Such operators have a description depending on a connection on the bundle and an endomorphism of the bundle as in [39, Lemma 1.2.1], and in our situation, for *Eρ*, the bundle connection is flat (curvature ≡ 0) and the endomorphism *e* is trivial.

The coefficients *uk(ρ)(x)* (as a function of *x* ∈ *M*) are of the form

$$
\mu\_k(\Delta\_\rho)(\mathbf{x}) = \text{tr}\_{E\_{\rho,\mathbf{x}}}(e\_k(\Delta\_\rho)(\mathbf{x})),
$$

where tr*Eρ ,x* denotes the fiberwise trace in the fibers *Eρ ,x* ∼= **<sup>C</sup>***<sup>N</sup>* , and where *ek(ρ)* is a linear combination with universal coefficients (independent of the dimension *n* of the manifold and the rank *N* of the bundle) of covariant derivatives of *R* · 1*<sup>N</sup>* (where *R* is the covariant Riemann curvature tensor of *M*), the bundle curvature and the endomorphism *e* [39, §3.1.8–3.1.9]. Since the latter two are identically zero in our situation, we can write *ek(ρ)* = *Pk* · 1*<sup>N</sup>* with *Pk* only depending on the covariant derivatives of *R*, in particular, not depending on *ρ*. We conclude that

$$(4\pi)^{-n/2} \int\_M u\_{\hbar/2}(\Delta\_\rho)(\mathbf{x}) = (4\pi)^{-n/2} \int\_M \text{tr}\_{E\_{\rho,\mathbf{x}}}(P\_{\hbar/2}(\mathbf{x}) \cdot \mathbf{l}\_N) = NU,$$

where *<sup>U</sup>* <sup>=</sup> *(*4*π )*−*n/*<sup>2</sup> *<sup>M</sup> Pn/*<sup>2</sup> is independent of *ρ*.

Therefore, applying (3.8) to and *ρ*, we find

$$\dim \ker \Delta\_{\rho} = NU - \zeta\_{\Delta\_{\rho}}(0) = N(\zeta\_{\Delta}(0) + 1) - \zeta\_{\Delta\_{\rho}}(0). \tag{3.9}$$

We can compute the rank *N* in terms of the first coefficients in the asymptotic expansions: using *e*0*(ρ)* = 1*<sup>N</sup>* , we find *N* = *<sup>M</sup> u*0*(ρ)/ <sup>M</sup> u*0*()*. On the other hand, *ζρ (s)* has a simple pole at *<sup>s</sup>* <sup>=</sup> *n/*<sup>2</sup> with residue *(n/*2*)*−<sup>1</sup> *<sup>M</sup> u*0*(ρ)* (see, e.g., the proof of [83, Thm. 5.2]), so that the function *ζρ (s)/ζ(s)* is holomorphic at *s* = *n/*2 and takes value *N* there:

$$N = \left. \frac{\xi\_{\Delta\_{\rho}}(s)}{\xi\_{\Delta}(s)} \right|\_{s=\frac{n}{2}}.\tag{3.10}$$

Combining Eqs. (3.9) and (3.10) gives the desired expression (3.7) for the multiplicity of zero in the spectrum in terms of spectral zeta functions only. 

### *Remark 3.10.3*

(i) The above argument also shows that for a twisted Laplacian *ρ* corresponding to a unitary representation on a fixed manifold *M*, the value

$$\dim \ker \Delta\_{\rho} + \zeta\_{\Delta\_{\rho}}(0)$$

only depends on the dimension of the representation *ρ*.

(ii) Weyl's law for *ρ* says that if N*(ρ, X)* denotes its number of eigenvalues ≤ *X*, then

$$\lim\_{X \to +\infty} \frac{\mathcal{N}(\Delta\_{\rho}, X)}{X^{n/2}} = N \cdot \frac{\text{vol}(M)}{(4\pi)^{n/2} \Gamma\left(\frac{n}{2} + 1\right)},$$

so that on a fixed manifold, the dimension *N* of the representation *ρ* can be read off from the asymptotics of the spectra of *ρ* and *M*:

$$N = \lim\_{X \to +\infty} \frac{\mathbf{N}(\Delta\_{\rho}, X)}{\mathbf{N}(\Delta\_{M}, X)};\tag{3.11}$$

formulas (3.10) and (3.11) are equivalent through Karamata's version of the Tauberian theorem (compare [11, pp. 91–92]).

### **Open Problem**

Find a "geometric" formula for the difference in multiplicities of zero for two operators *A* and *B* on a manifold of the general type considered here that have identical non-zero spectrum.

### **Open Problem**

Is the multiplicity of zero in the spectrum of a twisted Laplacian determined by the non-zero spectrum of the twisted Laplacian alone, without assuming knowledge of the spectrum of the usual Laplacian?

### **Open Problem**

Study how *disjoint* the spectra of the different *ρi* in (3.6) are, similar to the question how disjoint zeros of number theoretic *L*-series are (cf. [84]): the socalled *grand simplicity hypothesis* says that the imaginary parts of the zeros of all Dirichlet *L*-series for primitive characters are linearly independent over **Q**. From the above decomposition results, it is clear that if *ρ* is a subrepresentation of *ρ*, then *σ (ρ )* ⊆ *σ (ρ)* (this even holds for infinite amenable groups if *ρ* is weakly contained in *ρ*, cf. [92]); here, we are asking for a kind of converse result.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 4 Detecting Representation Isomorphism Through Twisted Spectra**

In this chapter, we give a spectral characterisation of isomorphism of induced representations. We also discuss strong isospectrality in the sense of Pesce (which, by a result of Sunada, is implied by weak conjugacy of subgroups), discuss an illustrative example of lens spaces due to Ikeda, and use the first result to give a spectral characterisation of weak conjugacy.

### **4.1 Spectral Detection of Isomorphism of Induced Representations**

In this section, we assume again that we have a diagram (1.2)

of finite coverings. We start with a proposition that allows us to detect isomorphism of representations induced from linear characters purely from spectral data. **Proposition 4.1.1** *For two linear characters* χ1 <sup>∈</sup> Hq1 *and* χ2 <sup>∈</sup> Hq2 *, the following* 

*are equivalent:* 


31


*Remark 4.1.2* Condition (ii') is

$$\begin{cases} \sigma\_{M\_1}(\overline{\chi}\_1 \otimes \mathrm{Res}\_{H\_1}^G \mathrm{Ind}\_{H\_1}^G \chi\_1) = \sigma\_{M\_2}(\overline{\chi}\_2 \otimes \mathrm{Res}\_{H\_2}^G \mathrm{Ind}\_{H\_1}^G \chi\_1); \\ \sigma\_{M\_1}(\overline{\chi}\_1 \otimes \mathrm{Res}\_{H\_1}^G \mathrm{Ind}\_{H\_2}^G \chi\_2) = \sigma\_{M\_2}(\overline{\chi}\_2 \otimes \mathrm{Res}\_{H\_2}^G \mathrm{Ind}\_{H\_2}^G \chi\_2). \end{cases}$$

In this form, the statement of the proposition is similar to a number-theoretical result of Solomatin [89] that inspired our proof below.

*Proof of Proposition 4.1.1* We start by proving that (i) implies (ii). Let ρ denote any irreducible representation of G; then for any i <sup>=</sup> <sup>1</sup>, 2, we have

$$\begin{split} \langle \mathrm{Ind}\_{H\_{l}}^{G} \left( \overline{\chi}\_{l} \otimes \mathrm{Res}\_{H\_{l}}^{G} \mathrm{Ind}\_{H\_{l}}^{G} \chi\_{l} \right), \rho \rangle &= \langle \overline{\chi}\_{l} \otimes \mathrm{Res}\_{H\_{l}}^{G} \mathrm{Ind}\_{H\_{l}}^{G} \chi\_{l}, \mathrm{Res}\_{H\_{l}}^{G} \rho \rangle \\ &= \langle \overline{\chi}\_{l}, \mathrm{Res}\_{H\_{l}}^{G} \rho \otimes \overline{\mathrm{Res}\_{H\_{l}}^{G} \mathrm{Ind}\_{H\_{l}}^{G} \chi\_{l}} \rangle \\ &= \langle \overline{\chi}\_{l}, \mathrm{Res}\_{H\_{l}}^{G} \left( \rho \otimes \overline{\mathrm{Ind}\_{H\_{l}}^{G} \chi\_{l}} \right) \rangle \\ &= \langle \overline{\mathrm{Ind}\_{H\_{l}}^{G} \chi\_{l}}, \rho \otimes \overline{\mathrm{Ind}\_{H\_{l}}^{G} \chi\_{l}} \rangle. \end{split}$$

By assumption (i), this final expression is independent of i <sup>=</sup> <sup>1</sup>, 2, and hence the same holds for the initial expression. Since this holds for any ρ, we find that

$$\operatorname{Ind}\_{H\_1}^G \left( \overline{\chi}\_1 \otimes \operatorname{Res}\_{H\_1}^G \operatorname{Ind}\_{H\_1}^G \chi\_1 \right) = \operatorname{Ind}\_{H\_2}^G \left( \overline{\chi}\_2 \otimes \operatorname{Res}\_{H\_2}^G \operatorname{Ind}\_{H\_2}^G \chi\_2 \right).$$

By Lemma 3.8.1, we find

$$\begin{aligned} \sigma\_{M\_1}(\overline{\chi}\_1 \otimes \mathrm{Res}\_{H\_1}^G \mathrm{Ind}\_{H\_1}^G \chi\_1) &= \sigma\_{M\_2}(\overline{\chi}\_2 \otimes \mathrm{Res}\_{H\_2}^G \mathrm{Ind}\_{H\_2}^G \chi\_2) \\ &= \sigma\_{M\_2}(\overline{\chi}\_2 \otimes \mathrm{Res}\_{H\_2}^G \mathrm{Ind}\_{H\_1}^G \chi\_1), \end{aligned} \tag{4.1}$$

the last line again by assumption (i). This is condition (ii) for (i, j ) <sup>=</sup> (1, <sup>1</sup>) and (i, j ) <sup>=</sup> (2, <sup>1</sup>). Using assumption (i), one may replace Ind<sup>G</sup> <sup>H</sup><sup>1</sup> <sup>χ</sup><sup>1</sup> by Ind<sup>G</sup> <sup>H</sup><sup>2</sup> <sup>χ</sup><sup>2</sup> in formula (4.1) on one or both sides, and this shows condition (ii) for all other choices of i, j .

Passing from stronger to weaker statements, (ii) implies (ii') and (iii), and (iii), as well as (ii'), imply (iii'). Hence we only need to prove that (iii') implies (i). Consider, for different i, j ,

$$a\_{l,j} := \langle \overline{\chi}\_l \otimes \text{Res}\_{H\_l}^G \text{Ind}\_{H\_j}^G \chi\_j, \mathbf{1} \rangle = \langle \text{Ind}\_{H\_j}^G \chi\_j, \text{Ind}\_{H\_l}^G \chi\_l \rangle,\tag{4.2}$$

where the last equality follows by Frobenius reciprocity. Setting ψ to be the class function ψ := Ind<sup>G</sup> <sup>H</sup><sup>1</sup> <sup>χ</sup><sup>1</sup> <sup>−</sup> Ind<sup>G</sup> <sup>H</sup><sup>2</sup> <sup>χ</sup>2, this allows us to compute that

$$
\langle \psi, \psi \rangle = a\_{\mathbb{I}, \mathbb{I}} + a\_{2,2} - a\_{\mathbb{I},2} - a\_{2,\mathbb{I}},
$$

and since in (iii') we are assuming <sup>a</sup>1,<sup>1</sup> <sup>=</sup> <sup>a</sup>2,<sup>1</sup> and <sup>a</sup>1,<sup>2</sup> <sup>=</sup> <sup>a</sup>2,2, it follows that ψ, ψ = 0, so ψ <sup>=</sup> 0, which is condition (i).

### **4.2 Strong Isospectrality and Spectral Detection of Weak Conjugacy**

Isospectrality of manifolds <sup>M</sup><sup>1</sup> and <sup>M</sup><sup>2</sup> in a diagram of the form (1.2) does not in general imply that <sup>H</sup><sup>1</sup> and <sup>H</sup><sup>2</sup> are weakly conjugate.

*Example 4.2.1* Consider the situation where <sup>M</sup> <sup>=</sup> <sup>S</sup>5, and <sup>M</sup><sup>1</sup> <sup>=</sup> L(11; <sup>1</sup>, <sup>2</sup>, <sup>3</sup>) and <sup>M</sup><sup>2</sup> <sup>=</sup> L(11; <sup>1</sup>, <sup>2</sup>, <sup>4</sup>) are lens spaces L(q;s1, s2, s3) defined as the quotient of S<sup>5</sup> by the block diagonal 6 × 6 matrix given by three 2 × 2 blocks representing planar rotations over respective angles <sup>2</sup>π si/q. In this case, the two groups Hi ∼= **<sup>Z</sup>**/11**<sup>Z</sup>** are not equal as subgroups of the isometry group of S5; they commute, and we can set

$$M\_0 = (H\_1 \times H\_2) \backslash M.$$

Ikeda has shown that <sup>M</sup><sup>1</sup> and <sup>M</sup><sup>2</sup> are isospectral for the Laplace operator on functions (see [52, p. 313], observing that since 8 = −3 mod 11, by Ikeda [52, Thm. 2.1] <sup>M</sup><sup>1</sup> is isometric to L(11; <sup>1</sup>, <sup>2</sup>, <sup>8</sup>), where the latter parameters are the ones used by Ikeda). However, <sup>H</sup><sup>1</sup> and <sup>H</sup><sup>2</sup> are not weakly conjugate: since <sup>G</sup> is abelian, conjugacy classes are singletons and |{g} ∩ Hi<sup>|</sup> is 0 or 1 depending on whether g is in Hi or not. ♦

Sunada [90, Lemma 1] proved that weak conjugacy of <sup>H</sup><sup>1</sup> and <sup>H</sup><sup>2</sup> implies *strong isospectrality*, defined as follows in the sense of Pesce [79, §II].

**Definition 4.2.2** Two Riemannian manifolds <sup>M</sup><sup>1</sup> and <sup>M</sup><sup>2</sup> admitting a common cover M are called *strongly isospectral* if the spectra of qi∗A acting on <sup>L</sup>2(Mi, qi∗E) ∼= <sup>L</sup>2(M, E)Hi are equal for any natural operator <sup>A</sup> on M. Here, qi : <sup>M</sup> Mi are the corresponding covering maps, and an operator <sup>A</sup> as above on M is *natural* if G acts isometrically on the fibers of the bundle E and A commutes with the action of G.

The Laplace operators acting on k-forms (see Example 3.1.2) are natural for all <sup>k</sup>, so if <sup>H</sup><sup>1</sup> and <sup>H</sup><sup>2</sup> are weakly conjugate, then the spectra of all of these are equal (sometimes, "strong isospectrality" is used to mean that precisely these operators are isospectral, but we will follow Pesce's definition as above).

*Example 4.2.3* The lens spaces in Example 4.2.1 are isospectral for the Laplacian on functions, but not on all k-forms, cf. [60, Remark 3.8]. In fact, <sup>M</sup><sup>1</sup> and <sup>M</sup><sup>2</sup> from Example 4.2.1 are isospectral on functions, but not on 1-forms, as is explained in [53, Example 1, p. 416] (in the notation of loc. cit., <sup>M</sup><sup>1</sup> <sup>=</sup> <sup>L</sup><sup>2</sup> and <sup>M</sup><sup>2</sup> is isometric to <sup>L</sup><sup>1</sup> <sup>=</sup> L(11; <sup>1</sup>, <sup>2</sup>, <sup>5</sup>)).

Strongly isospectral lens spaces are isometric [60, Proposition 7.2]. There exist lens spaces isospectral on k-forms for all k, while at the same time not strongly isospectral (i.e., isometric) in the sense of Definition 4.2.2; for example, M<sup>1</sup> <sup>=</sup> L(49; <sup>1</sup>, <sup>6</sup>, <sup>15</sup>) and <sup>M</sup><sup>2</sup> <sup>=</sup> L(49; <sup>1</sup>, <sup>6</sup>, <sup>20</sup>). This follows from the characterisation of k-isospectrality for all k in terms of lattice norms with an extra geometric condition in [60], which permits the construction of many examples, as well as infinite families of pairs of lens spaces with this property, e.g., [60, Table 1 and Theorem 7.1]. ♦

The next Proposition 4.2.4 provides a spectral criterion that is equivalent to weak conjugacy, and is an immediate corollary of Proposition 4.1.1. It is analogous to a number theoretical result of Nagata [73].

**Proposition 4.2.4** *Suppose* M *is a connected smooth closed Riemannian manifold,*  <sup>G</sup> *a finite group of isometries of* <sup>M</sup> *and* <sup>H</sup><sup>1</sup> *and* <sup>H</sup><sup>2</sup> *are two subgroups of fixedpoint free isometries in* <sup>G</sup> *with associated quotient manifolds* <sup>M</sup><sup>1</sup> := <sup>H</sup>1\<sup>M</sup> *and*  <sup>M</sup><sup>2</sup> := <sup>H</sup>2\M*. Then* <sup>H</sup><sup>1</sup> *and* <sup>H</sup><sup>2</sup> *are weakly conjugate if and only if the multiplicity of the zero eigenvalue in* σMi(Res<sup>G</sup> Hi Ind<sup>G</sup> **<sup>1</sup>**) *is independent of* i, j <sup>=</sup> <sup>1</sup>, <sup>2</sup>*.* 

*Proof* The groups <sup>H</sup><sup>1</sup> and <sup>H</sup><sup>2</sup> are weakly conjugate precisely when there is an isomorphism of permutation representations Ind<sup>G</sup> <sup>H</sup><sup>1</sup> **<sup>1</sup>** ∼= Ind<sup>G</sup> <sup>H</sup><sup>2</sup> **<sup>1</sup>**. By Proposition 4.1.1, this is equivalent to the claim.

Hj

One may vary the condition of twisted isospectrality using the equivalent conditions in Proposition 4.1.1. For example, using Remark 4.1.2, we deduce the following (adding some redundant information).

**Corollary 4.2.5** *If a diagram as in* (1.2) *is given, and the following twisted spectra agree:* 

$$\begin{cases} \sigma\_{M\_1}(\operatorname{Res}\_{H\_1}^G \operatorname{Ind}\_{H\_1}^G \mathbf{1}) = \sigma\_{M\_2}(\operatorname{Res}\_{H\_2}^G \operatorname{Ind}\_{H\_1}^G \mathbf{1});\\ \sigma\_{M\_1}(\operatorname{Res}\_{H\_1}^G \operatorname{Ind}\_{H\_2}^G \mathbf{1}) = \sigma\_{M\_2}(\operatorname{Res}\_{H\_2}^G \operatorname{Ind}\_{H\_2}^G \mathbf{1}), \end{cases}$$

*then the manifolds* <sup>M</sup><sup>1</sup> *and* <sup>M</sup><sup>2</sup> *are strongly isospectral.* 

*Remark 4.2.6* Mackey's theorem describes how, for two subgroups <sup>K</sup><sup>1</sup> and <sup>K</sup><sup>2</sup> of a group G, a representation of the form Res<sup>G</sup> <sup>K</sup><sup>2</sup> Ind<sup>G</sup> <sup>K</sup><sup>1</sup> <sup>ρ</sup> splits into irreducibles (see, e.g., [88, Prop. 22]). In the situation of Proposition 4.2.4, with Ki ∈ {H1, H2}, we find that Res<sup>G</sup> <sup>K</sup><sup>2</sup> Ind<sup>G</sup> <sup>K</sup><sup>1</sup> **<sup>1</sup>** splits as the direct sum of the permutation representations corresponding to the action of <sup>K</sup><sup>2</sup> on the cosets of sK1s−<sup>1</sup> <sup>∩</sup> <sup>K</sup><sup>2</sup> for <sup>s</sup> <sup>∈</sup> <sup>K</sup>2\G/K1, and the occurring spectra are the (multiset-)union of the spectra corresponding to these representations; for example,

$$\sigma\_{M\_l}(\operatorname{Res}\_{H\_l}^G \operatorname{Ind}\_{H\_l}^G \mathbf{1}) = \bigcup\_{s \in H \backslash G/H\_l} \sigma\_{M\_l}(\operatorname{Ind}\_{sH\_l s^{-1} \cap H\_l}^{H\_l} \mathbf{1}),$$

contains the usual Laplace spectra σMi(Mi) (setting s to be the trivial double coset).

### **Project**

It is possible that, given an integer <sup>p</sup><sup>0</sup> satisfying <sup>0</sup> < p<sup>0</sup> <sup>≤</sup> <sup>n</sup>, n-dimensional manifolds are isospectral for the Laplacian on p-forms for all p<p0, but not for p <sup>=</sup> p0, and one could say that the larger p0/n, the "more strongly isospectral" the manifolds are. Can one give a geometric meaning to p0/n, given two manifolds (possibly of special type)? For example, this problem is solved by Lauret [59] for lens spaces by encoding some geometric properties in a generating series.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 5 Representations with a Unique Monomial Structure**

In this chapter, we recall the notion of monomial structures (and their isomorphism) on a representation, show a natural monomial structure on induced representations, and introduce solitary characters (characters whose induced representation has a unique monomial structure up to isomorphism); these characters may be used to detect conjugacy of subgroups. We also recall a specific type of wreath product construction and state and prove Bart de Smit's theorem on the existence of solitary characters for these (and a follow-up result of Pintonello for characters of degree two)—these were previously formulated and used in the context of number theory, but we present them abstractly. We give an application to covering equivalence in a very specific setup of manifolds, and also count the number of required characters, based on a formula for the commutator of a wreath product.

### **5.1 Monomial Structures**

**Definition 5.1.1** Suppose *ρ* : *G* → Aut*(V )* is a representation, and

$$V = \bigoplus\_{x \in \Omega} \mathcal{E}\_x$$

is a decomposition of *V* into one-dimensional spaces ("lines") L*<sup>x</sup>* for *x* ∈ , with some index set. If the action of *G* on *V* permutes the lines L*<sup>x</sup>* , we say that the *G*-set

$$L = \{ \mathcal{Q}\_{\boldsymbol{\lambda}} \colon \boldsymbol{x} \in \Omega \}$$

is a *monomial structure* on *ρ*.

Equivalently, in a basis having precisely one element from each line L*<sup>x</sup>* , the action of any *g* ∈ *G* is given by a matrix having exactly one non-zero entry in each row and column. Note that, contrary to the case of permutation matrices, the non-zero entry in the matrix need not be 1.

An *isomorphism of monomial structures L* and *L* on two representation of the same group *G* is an isomorphism of *L* and *L* as *G*-sets.

*Example 5.1.2* An induced representation Ind*<sup>G</sup> <sup>H</sup> <sup>χ</sup>* of a linear character *<sup>χ</sup>* <sup>∈</sup> *<sup>H</sup>*<sup>q</sup> admits (by definition) a monomial structure where = {*g*1*,...,gn*} is such that *giH* are the different cosets of *H* in *G*, and L*<sup>x</sup>* = **C** · *xH*. The corresponding matrices have as non-zero entries *n*-th roots of unity if *χ* is a character of order *n*. We call this monomial structure the *standard monomial structure* on Ind*<sup>G</sup> <sup>H</sup> χ*. This standard monomial structure is isomorphic to *G/H* as *G*-set. ♦

**Definition 5.1.3** A linear character on a subgroup *H* of a group *G* is called *Gsolitary* if Ind*<sup>G</sup> <sup>H</sup>* has a unique monomial structure up to isomorphism.

**Lemma 5.1.4** *Let G denote a group with two subgroups H*1 *and H*2*, and suppose* ∈ *H* <sup>|</sup>1 *is a G-solitary linear character. There exists a linear character <sup>χ</sup>* <sup>∈</sup> *<sup>H</sup>* <sup>|</sup>2 *for which there is an isomorphism of representations* Ind*<sup>G</sup> <sup>H</sup>*<sup>1</sup> ∼= Ind*<sup>G</sup> <sup>H</sup>*2 *χ if and only if H*1 *and H*2 *are conjugate subgroups of G.* 

*Proof* In this situation, Ind*<sup>G</sup> <sup>H</sup>*2 *χ* carries two monomial structures: the standard one and the one induced from the standard one on Ind*<sup>G</sup> <sup>H</sup>*<sup>1</sup> through the isomorphism of representations. Hence these monomial structures have to be isomorphic. But as *G*-sets, they are *G/H*1 and *G/H*2, respectively (see Example 5.1.2). By Proposition 3.5.1(ii), this means precisely that *H*1 and *H*2 are conjugate in *G*.

### **5.2 Wreath Product Construction**

**Definition 5.2.1** Let *G* denote a finite group and *H* a subgroup of index *n* := [*G* : *H*] with cosets

$$\{\mathbf{g}\_1 H = H, \mathbf{g}\_2 H, \dots, \mathbf{g}\_n H\}$$

of cardinality *n*. For a prime number , let *C* = **Z***/***Z** denote the cyclic group with elements, and let

$$
\tilde{G} := C^n \rtimes G
$$

denote the *wreath product*; this is by definition the semidirect product where *G* acts on the *n* copies of *C* by permuting the coordinates in the same way as *G* permutes the cosets *giH*. In coordinates, this means that if we let *e*1*,...,en* denote the standard basis vectors of *Cn*, and, as before, define the permutation *<sup>i</sup>* → *g(i)* of {1*,...,n*} by *ggiH* = *gg(i)H*, then the semidirect product is defined by the action

$$G \stackrel{\Phi}{\rightarrow} \text{Aut}(C'') \colon g \mapsto \Phi(g) = \left[ \sum\_{j=1}^{n} k\_j e\_j \mapsto \sum\_{j=1}^{n} k\_j e\_{\mathfrak{g}(j)} \right] \tag{5.1}$$

where *kj* <sup>∈</sup> **<sup>Z</sup>***/***Z**. This is the (left) action of *<sup>g</sup>* <sup>∈</sup> *<sup>G</sup>* on *C<sup>n</sup>* given by

$$C^n \ni (k\_1, \ldots, k\_n) \mapsto (k\_{\mathfrak{g}^{-1}(1)}, \ldots, k\_{\mathfrak{g}^{-1}(n)}) \in C^n.$$

Define

*<sup>H</sup>* := *<sup>C</sup><sup>n</sup>* -*H*

to be the subgroup of *<sup>G</sup>* corresponding to *H*. The cosets of *<sup>H</sup>* in *<sup>G</sup>* are of the form

$$\{\widetilde{\mathbf{g}}\_1\widetilde{H} = \widetilde{H}, \widetilde{\mathbf{g}}\_2\widetilde{H}, \dots, \widetilde{\mathbf{g}}\_n\widetilde{H}\},$$

where for *gi* <sup>∈</sup> *<sup>G</sup>*, we have a corresponding element *gi* := *(*0*, gi)* <sup>∈</sup> *<sup>G</sup>*.

*Remark 5.2.2* Recall that Ind*<sup>G</sup> <sup>H</sup>* **1** is the **Z**[*G*]-module corresponding to the permutation representation of *G* acting on the *G*-cosets of *H*. Thus, if we identify *C* with the additive group of the finite field **F**, the action of *<sup>G</sup>* on *C<sup>n</sup>* ∼= **<sup>F</sup>***<sup>n</sup>* corresponds to the **F**[*G*]-module *(*Ind*<sup>H</sup> <sup>G</sup>* **1***)* ⊗**Z F**.

**Proposition 5.2.3 (Bart de Smit [28, §10])** *For all* <sup>≥</sup> <sup>3</sup>*, there exists a <sup>G</sup>-solitary character of order on <sup>H</sup>.* 

*Proof* Define by

$$\Xi \colon \widetilde{H} \to \mathbf{C}^\* \colon (k\_1, \dots, k\_n, \mathbf{g}) \mapsto e^{2\pi i k\_1/\ell} \,. \tag{5.2}$$

Let *L* = {L*<sup>x</sup>* } and *L* = {L *<sup>x</sup>* } denote two monomial structures on *<sup>ρ</sup>* := Ind*G <sup>H</sup>* , where *<sup>L</sup>* is the standard one (see Example 5.1.2). The action of *<sup>G</sup>* <sup>≤</sup> *<sup>G</sup>* on *<sup>L</sup>* is that of *<sup>G</sup>* on *G/H* and (after rearranging) the action of *C<sup>n</sup>* <sup>≤</sup> *<sup>G</sup>* is given by

$$((k\_1, \ldots, k\_n) \cdot \mathcal{A}\_{\sf J}^{\rho} = e^{2\pi ik\_{\sf j}/\ell} \cdot \mathcal{A}\_{\sf j},\tag{5.3}$$

where we used the simplified notation L*<sup>j</sup>* := <sup>L</sup>*gjH*. The character *<sup>ψ</sup>* of *<sup>ρ</sup>* can be computed using as basis any set of vectors from the lines in *L* or *L* . From the above,

$$|\psi((1,0,\ldots,0))| = |e^{2\pi i/\ell} + \underbrace{1 + \cdots + 1}\_{n-1}| > n - 2, \ell$$

where the last inequality is strict since ≥ 3. On the other hand, computing the same trace using a basis from *L* , we get a sum of some number, say, *m*, of -th roots of unity, where *m* is the number of lines in *L* that are mapped to itself by *(*1*,* 0*,... ,* 0*)*. If there is a line not mapped to itself (a zero diagonal entry in the corresponding matrix), then there are at least two (since every row/column has precisely two nonzero entries), so *m* = *n* or *m* ≤ *n* − 2. In the latter case, |*ψ((*1*,* 0*,...,* 0*))*| ≤ *n* − 2*,* which is impossible. Since *C<sup>n</sup>* is generated by *G*-conjugates of *(*1*,* 0*,...,* 0*)*, we find that *C<sup>n</sup>* fixes all lines in *L* . Hence *L* ⊆ *L*, but since |*L*|=|*L* |=[*G* : *<sup>H</sup>*], we have *L* = *L* .

Pintonello [80, Theorem 3.2.2] has shown that for = 2, there does not always exist a solitary character as in Proposition 5.2.3. However, he also proved the following result, of which we give a self-contained proof.

**Proposition 5.2.4 (Pintonello [80, Theorem 2.3.1])** *Given a group G with two subgroups H*1 *and H*2*, consider the corresponding wreath products G, <sup>H</sup>*1 *and <sup>H</sup>*<sup>2</sup> *with <sup>C</sup>* <sup>=</sup> **<sup>Z</sup>***/*2**Z***. Set* : *<sup>H</sup>*1 <sup>→</sup> **<sup>C</sup>**<sup>∗</sup> : *(k*1*,...,kn, g)* → *(*−1*)k*<sup>1</sup> *, and assume that both* 

$$\operatorname{Ind}\_{\tilde{H}\_1}^{\tilde{G}} \mathbf{1} \cong \operatorname{Ind}\_{\tilde{H}\_2}^{\tilde{G}} \mathbf{1} \, and \tag{5.4}$$

$$\operatorname{Ind}\_{\tilde{H}\_1}^{\tilde{Q}} \Xi \cong \operatorname{Ind}\_{\tilde{H}\_2}^{\tilde{Q}} \chi,\tag{5.5}$$

*for some linear character <sup>χ</sup> on <sup>H</sup>*2*. Then <sup>H</sup>*1 *and <sup>H</sup>*2 *are conjugate in G.* 

*Proof* Equality (5.5) induces two monomial structures *L*1 and *L*2 on *<sup>ρ</sup>* := Ind*G H*1 , where *Li* is isomorphic to *G/ <sup>H</sup>i*. As in (5.3), *<sup>ε</sup>* := *(*1*,* <sup>0</sup>*,...,* <sup>0</sup>*)* <sup>∈</sup> *<sup>C</sup><sup>n</sup>* <sup>≤</sup> *<sup>G</sup>* fixes all lines in *L*1. Note that the number of lines in *Li* fixed by *ε* is the value of the character of Ind*G Hi* **1** at *ε*, given in (3.3), and by (5.4), these are equal for *i* = 1 and *i* = 2. Hence all lines in *L*2 are fixed by *ε*, and as in the previous proof, we conclude that *C<sup>n</sup>* fixes all lines in *L*2. Hence *L*2 <sup>⊆</sup> *<sup>L</sup>*1, but since <sup>|</sup>*L*1|=|*L*2|=[*G* : *<sup>H</sup>i*], we have *L*1 = *L*2.

### **5.3 Application to Manifolds**

We deduce the following intermediate result.

**Corollary 5.3.1** *Suppose we have a diagram* (1.2)*. Let C* := **Z***/***Z** *denote a cyclic group of prime order*  <sup>≥</sup> <sup>3</sup>*. Let G and H*<sup>1</sup> *denote the wreath products as in Definition* 5.2.1 (*with <sup>H</sup>* <sup>=</sup> *<sup>H</sup>*1) *and H*<sup>2</sup> := *<sup>C</sup><sup>n</sup>* - *H*<sup>2</sup> (*with the same action defined via the H*1*-cosets*)*, and assume that there exists a diagram of Riemannian coverings*

*Then M*<sup>1</sup> *and M*<sup>2</sup> *are equivalent Riemannian covers of M*<sup>0</sup> *if and only if for a G solitary character on <sup>H</sup>*<sup>1</sup> *and for some linear character <sup>χ</sup> on <sup>H</sup>*2*, the multiplicity of zero is equal in the two spectra* 

$$\sigma\_{M\_1}(\overline{\Xi}\otimes \mathrm{Res}\_{\widetilde{H}\_1}^{\widetilde{G}} \mathrm{Ind}\_{\widetilde{H}\_1}^{\widetilde{G}} \Xi) \text{ and } \sigma\_{M\_2}(\overline{\chi}\otimes \mathrm{Res}\_{\widetilde{H}\_2}^{\widetilde{G}} \mathrm{Ind}\_{\widetilde{H}\_1}^{\widetilde{G}} \Xi)$$

*and in the two spectra* 

$$
\sigma\_{M\_1}(\overline{\Xi} \otimes \mathrm{Res}\_{\widetilde{H}\_1}^{\widetilde{G}} \mathrm{Ind}\_{\widetilde{H}\_2}^{\widetilde{G}} \chi) \, and \,\sigma\_{M\_2}(\overline{\chi} \otimes \mathrm{Res}\_{\widetilde{H}\_2}^{\widetilde{G}} \mathrm{Ind}\_{\widetilde{H}\_2}^{\widetilde{G}} \chi) \,.
$$

*Proof* First of all, since <sup>≥</sup> 3,a *G*-solitary character on *H*<sup>1</sup> exists, by Proposition 5.2.3. By Proposition 4.1.1, the equalities of multiplicities of zero is equivalent to Ind*G H*1 ∼= Ind*G H*2 *<sup>χ</sup>*. Since is *G*-solitary, we conclude by Lemma 5.1.4 that *H*<sup>1</sup> and *H*<sup>2</sup> are conjugate in *G*. As *C<sup>n</sup>* is normal in *H*<sup>2</sup> with quotient *<sup>H</sup>*2, we find that *<sup>H</sup>*2\*M* <sup>=</sup> *<sup>H</sup>*2\*<sup>M</sup>* <sup>=</sup> *<sup>M</sup>*<sup>2</sup> and hence this conjugacy defines an isometry from *M*<sup>1</sup> to *M*<sup>2</sup> that is the identity on *M*0.

Since *<sup>χ</sup>* runs over linear characters of *H*2, the "less abelian" the extension is, the less spectra need to be compared. A more precise statement is the following, where we use the abelianisation *H*ab <sup>2</sup> of *H*2, defined as the quotient of *H*<sup>2</sup> by the subgroup generated by commutators (equivalently, the largest abelian quotient of *H*2; equivalently, *H*ab <sup>2</sup> ∼= Hom*(H*2*,* **C**∗*)*). The two extremes are then: if *H*<sup>2</sup> is abelian, *H*ab <sup>2</sup> is as large as *<sup>H</sup>*2; but if *<sup>H</sup>*<sup>2</sup> is non-abelian simple, then <sup>|</sup>*H*ab <sup>2</sup> | = 1.

**Proposition 5.3.2** *In the setup of Corollary* 5.3.1*, the dimension of the representations of which the spectra are being compared is the index* [*G* : *H*2]*. Furthermore, the number of spectral equalities to be checked in Corollary* 5.3.1 *by using all possible linear characters on <sup>H</sup>*<sup>2</sup> *is bounded above by* <sup>2</sup> · |*H*ab <sup>2</sup> |*.* 

In Corollary 5.3.1 and Proposition 5.3.2, one may interchange the roles of *H*<sup>1</sup> and *H*2, which could lead to tighter results.

*Proof* The dimension of the representations we are considering, as induced representations, is the index [*G* : *<sup>H</sup>*2]=[*<sup>G</sup>* : *<sup>H</sup>*2].

The spectral criterion in the proposition requires testing of 2 equalities of spectra for each linear character on *<sup>H</sup>*2, so there are at most <sup>2</sup>|*H*ab <sup>2</sup> | equalities to be checked.

The commutator subgroup of a wreath product *<sup>H</sup>*<sup>2</sup> <sup>=</sup> *<sup>C</sup>n*-*H*<sup>2</sup> is computed in [69, Cor. 4.9], and we find that in our case, with = {*g*1*,...,gn*} a set of representatives for the cosets,

$$|\langle \tilde{H}\_2, \tilde{H}\_2 \rangle| = |\langle H\_2, H\_2 \rangle| \cdot |\{ f \colon \Omega \to \mathcal{C} \colon \sum\_{\mathbf{y} \in \Omega} f(\mathbf{y}) = 0 \}|;$$

where, with |*C*| = , the second factor is ||−1. Hence we find |*H*ab <sup>2</sup> |=|*H*ab <sup>2</sup> | · *,* and the result follows.

*Remark 5.3.3* Using Proposition 4.1.1 to reformulate spectrally the extra assumption in Proposition 5.2.4 (where = 2), we find that in this case, the number of equalities to check is at most <sup>2</sup> <sup>+</sup> <sup>4</sup>|*H*ab <sup>2</sup> |.

*Remark 5.3.4* By Lemma 3.9.1, the multiplicity of zero in the spectrum *σM(ρ)* can be computed purely representation theoretically as the multiplicity of the trivial representation in *ρ*, which is in principle possible by Mackey theory (cf. Remark 4.2.6), but this would be going in reverse (from spectra to group theory instead of the other way around). Knowing the group *G* and its subgroups *H*<sup>1</sup> and *H*2, Riemannian equivalence of *M*<sup>1</sup> and *M*<sup>2</sup> over *M*<sup>0</sup> can be checked by a finite computation, verifying that *H*<sup>1</sup> and *H*<sup>2</sup> are conjugate in *G*. Corollary 5.3.1 translates this into a spectral statement (in the special setup where the group *G* is realised as indicated there).

*Remark 5.3.5* One may strip all geometric analysis from the results so far, and formulate the following purely group theoretical result. *Given a finite group G and two subgroups H*<sup>1</sup> *and H*2, *then* 

> *H*<sup>1</sup> and *H*<sup>2</sup> are conjugate in *G* if and only if Ind*G H*1 <sup>=</sup> Ind*G H*2 *χ*

*for some linear character <sup>χ</sup> on <sup>H</sup>*2. Here, *<sup>G</sup>* denotes the wreath product corresponding to the action of *G* on the *G*-cosets of *H*1, and denotes a solitary character of order 3 on *H*<sup>1</sup> (which exists by Proposition 5.2.3). The proof is immediate from Lemma 5.1.4 and the final sentence in the proof of Proposition 5.3.1. Observe that the construction of the wreath products and of is completely explicit, and the linear characters on *<sup>H</sup>*<sup>2</sup> can be described in terms of those on *<sup>H</sup>*<sup>2</sup> via the results used in the proof of Proposition 5.3.2.

In the next chapters, we study under which circumstances we have a cover as in Corollary 5.3.1, i.e., we deal with the realisation problem for the wreath product as isometry group of a cover, given an isometric free action of *G* on a closed manifold *M*. This is analogous to the inverse problem of Galois theory, realising the wreath product as Galois group of a number field. In manifolds, some condition is necessary on *M* for such an extension to be possible at all.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 6 Construction of Suitable Covers and Proof of the Main Theorem**

In this chapter, we set up some theory concerning the action of isometries on fundamental groups and first homology groups. More specifically, we show that the action of a finite group of isometries on the first homology group of a manifold corresponds to conjugation by a lift to the orbifold fundamental group of the quotient. This result is then used to give a necessary and sufficient condition for realising a certain wreath product by a Galois cover (by an explicit construction of the corresponding subgroup of the fundamental group)—a problem similar to the algebraic problem of inverse Galois theory. The required condition, called ( ∗), is a representation-theoretic property of the action on homology. Using the representation theoretical results from the previous chapter, this allows us to finish the proof of the main theorem of the text, describing covering equivalence of manifolds through spectra of twisted Laplacians.

### **6.1 Fundamental Group and First Homology**

We first fix some notations and constructions. Let M denote a connected closed oriented smooth Riemannian manifold. Fixing a point x <sup>∈</sup> M, the universal covering M is described as the set of homotopy classes [w] of paths w : [0, <sup>1</sup>] → M with w(0) <sup>=</sup> x. This provides a projection map : (M, --

$$
\Pi : (\bar{M}, \widetilde{\chi}) \to (M, \ge), \ \Pi([w]) = w(1),
$$

where x ∈ M represents the homotopy class of the constant path at x. If we equip M with the pull-back of the Riemannian metric on M then the group acts isometrically by deck transformations on M-, and M is identified with the quotient \M-.

Let <sup>∗</sup> denotes path-concatenation read from left to right, that is, [a]∗[b] is the homotopy class of the path obtained by first traversing a and then b. Letting x denote the constant path at x, we have an identification -

$$
\Phi\_{\widetilde{\mathbb{X}}} \colon \Gamma \to \pi\_1(M, \mathfrak{x})
$$

via the map x (γ ) = γ (x), with γ (x) representing a homotopy class of a closed loop starting and ending at x. More generally, any homotopy class of a path [w] in M induces a map [w] : <sup>→</sup> <sup>π</sup>1(M, w(1)) via [w](γ ) = [w−1] ∗ -

$$\Phi\_{[w]}(\boldsymbol{\gamma}) = [w^{-1}] \* \Phi\_{\widetilde{\boldsymbol{x}}}(\boldsymbol{\gamma}) \* [w]. \tag{6.1}$$

We denote the first homology group of M (with integer coefficients) by H1(M) <sup>=</sup> H1(M,**Z**). The universal coefficient theorem for homology [48, §3.A] implies that for any field K, we have an isomorphism

$$\mathcal{H}\_{\mathbb{L}}(M, K) = \mathcal{H}\_{\mathbb{L}}(M) \otimes\_{\mathbb{Z}} K...$$

Also, since M is a connected manifold, it is path-connected, so that we have a Hurewicz homomorphism inducing an identification

$$\mathcal{H}\_{\mathbb{I}}(M) = \pi\_{\mathbb{I}}(M, x)^{\text{ab}} \cong \Gamma^{\text{ab}}.$$

The map is given by considering a (homotopy class of a) loop as a concatenation of oriented 1-cells and mapping it to the (homology class of the) signed sum of those cells.

Composing the maps, we have a homomorphism <sup>0</sup> : <sup>→</sup> H1(M, **<sup>F</sup>**) given as the composition of [w] with the abelianisation map and the Hurewicz isomorphism, followed by reduction modulo :

$$\Gamma \xrightarrow{\Phi\_{\left[w\right]}\atop\mathbf{H}\_{\mathbf{I}}(M,w(1))} \xrightarrow{\cdot\_{\Psi}} \mathbf{H}\_{\mathbf{I}}(M,\mathbf{Z}) \xrightarrow{\otimes \mathbf{F}\_{\ell}} \mathbf{H}\_{\mathbf{I}}(M,\mathbf{F}\_{\ell}).\tag{6.2}$$

Since by (6.1) the homotopy classes of loops [w] for different <sup>w</sup> are freely homotopic, the composed map <sup>0</sup> is independent of the choice of w, as notation indicates. The standard choice is <sup>w</sup> <sup>=</sup> x, but we will naturally encounter others.

### **6.2 First Homology and Galois Covers**

Suppose now that <sup>G</sup> is a finite group of isometries acting on M, and let <sup>q</sup> : <sup>M</sup> <sup>→</sup> <sup>M</sup><sup>0</sup> denote the orbifold quotient, with <sup>0</sup> the covering group of -0 : M-<sup>→</sup> M0. Let be the normal subgroup of <sup>0</sup> corresponding to -: M- <sup>→</sup> M, so that there is a short exact sequence of groups

$$1 \to \Gamma \to \Gamma\_0 \xrightarrow{F} G \to 1 \tag{6.3}$$

giving an isomorphism <sup>G</sup> ∼= 0/ . The setup is summarised in diagram (6.4).

*M* 0 0 *M q G M*0 ∏ ∏Γ Γ (6.4)

**Definition 6.2.1** For <sup>γ</sup><sup>0</sup> <sup>∈</sup> 0, let conjγ<sup>0</sup> : <sup>→</sup> be conjugation by γ0, that is

$$\text{conj}\_{\mathbb{N}}(\boldsymbol{\gamma}) = \boldsymbol{\gamma}\_0 \boldsymbol{\gamma} \boldsymbol{\gamma}\_0^{-1}.$$

*Remark 6.2.2* If <sup>g</sup> <sup>∈</sup> <sup>G</sup> ∼= 0/ satisfies <sup>g</sup> <sup>=</sup> <sup>γ</sup>0, this represents the usual map

$$G \to \text{Out}(\Gamma) \colon \mathfrak{g} \mapsto \text{conj}\_{\mathbb{N}^{\mathbb{N}}}$$

induced by the exact sequence (6.3), where Out() <sup>=</sup> Aut()/Inn() is the quotient of the group of automorphisms of by the group Inn() = {conjγ : <sup>γ</sup> <sup>∈</sup> } of inner automorphisms.

In this setup, the action of the isometry <sup>g</sup> on M corresponds to the action of conjγ −1 0 on , so that there is a commuting diagram

$$\sum\_{M} \xleftarrow{\tilde{M}}\_{g} \xleftarrow{\nu\_{0}^{-1} \Gamma \gamma\_{0}} M} M \tag{6.5}$$

The action of G on M by isometries induces a linear action of G on H1(M, **<sup>F</sup>**), providing an **<sup>F</sup>**[G]-module structure on this homology group. The following lemma describes the relation between this action and the above outer conjugation on : they commute by the Hurewicz map. The argument is similar to the one used in the proof of Hopf's formula [21, (5.3)].

**Lemma 6.2.3** *Let* <sup>γ</sup><sup>0</sup> <sup>∈</sup> <sup>0</sup> *and* <sup>g</sup> <sup>∈</sup> <sup>G</sup> ∼= 0/ *such that* <sup>g</sup> <sup>=</sup> <sup>γ</sup>0*. Then the following diagram commutes:*

$$
\begin{array}{c}
\Gamma \xrightarrow[\text{W}\_{0}]{\text{conj}\_{\mathbb{V}\_{0}}} \Gamma \\
\Psi\_{0} \xleftarrow{\text{conj}\_{\mathbb{V}\_{0}}} \operatorname{\uph}\_{0} \\
\operatorname{H}\_{1}(M,\mathbf{F}\_{\ell}) \xrightarrow{\operatorname{g}\cdot} \operatorname{H}\_{1}(M,\mathbf{F}\_{\ell})
\end{array}
(1.6)
$$

*where* <sup>0</sup> *is as in* (6.2) *and the bottom line indicates the action of* <sup>g</sup> <sup>∈</sup> <sup>G</sup> *on the first homology group.* 

*Proof* The vertical maps are given by picking a base point x <sup>∈</sup> M, considering γ <sup>∈</sup> as a homotopy class of a closed loop in M based at x via x , rewriting <sup>γ</sup> as a concatenation of oriented 1-cells, and mapping these to the corresponding sum of 1-cells in homology. The crucial observation that makes the proof work is that we can decompose into 1-cells, not just 1-cycles, and the image is independent of the choice of base point. In this way, the closed loop corresponding to conj<sup>γ</sup><sup>0</sup> (γ ) decomposes as a (left-to-right) concatenation of the 1-cells

$$(\left(\wp\_0: \mathbf{x} \to \mathbf{g}\mathbf{x}\right)\*(\mathbf{g}\cdot\boldsymbol{\chi}:\mathbf{g}\boldsymbol{\chi} \to \mathbf{g}\boldsymbol{\chi})\*(\left(\boldsymbol{\chi}\_0^{-1}:\mathbf{g}\boldsymbol{\chi} \to \mathbf{x}\right),$$

that is mapped by <sup>0</sup> to the sum of homology classes [γ0] +<sup>g</sup> · [<sup>γ</sup> ]−[γ0] = <sup>g</sup> · [<sup>γ</sup> ], proving the commutativity of the diagram. 

*Remark 6.2.4* In fact, there is a larger commutative diagram:

$$\Psi\_{\vec{\pi}} \left( \begin{array}{c} \text{conj}\_{\mathbb{V}} \\ \begin{array}{c} \pi\_{1}(M,\mathtt{x}) \xrightarrow{\mathcal{g}\cdot} \pi\_{1}(M,\mathtt{g}\mathtt{x}) \\ \downarrow \\ \end{array} \right) \Psi\_{0} \\ \begin{array}{c} \text{for}\_{1}(M,\mathtt{F}\_{\ell}) \xrightarrow{\mathcal{g}\cdot} \text{H}\_{1}(M,\mathtt{F}\_{\ell}) \end{array} \right) \Psi\_{0} \end{array} \tag{6.7}$$

The detailed proof goes as follows: we have argued before that both vertical maps (from top to bottom) on the left and right hand side of (6.7) agree and are equal to 0. Naturality of the Hurewicz isomorphism guarantees commutativity of the lower square in (6.7) and it therefore suffices to prove commutativity of the upper square in (6.7). isometries on M-

Elements <sup>g</sup> <sup>∈</sup> <sup>G</sup> are isometries <sup>g</sup> : <sup>M</sup> <sup>→</sup> <sup>M</sup> and the groups <sup>0</sup> and acting by can be described via deck transformations as follows := {I : M- → Misometry <sup>|</sup> F (I ([w])) <sup>=</sup> F ([w]) ∀ [w] ∈ M-

$$\begin{aligned} &\text{metrics on } \mathcal{M} \text{ can be described via deck transformations as follows} \\ &\Gamma := \{ I : \widetilde{\mathcal{M}} \to \widetilde{\mathcal{M}} \text{ isometry} \mid F(I(\{w\})) = F(\{w\}) \,\forall \, [w] \in \widetilde{\mathcal{M}} \}, \\ &\Gamma\_0 := \{ I : \widetilde{\mathcal{M}} \to \widetilde{\mathcal{M}} \text{ isometry} \mid \exists \, \mathcal{g} \in G : F(I(\{w\})) = \operatorname{g} F(\{w\}) \,\forall \, [w] \in \widetilde{\mathcal{M}} \}. \end{aligned} \tag{6.8}$$

In this description, the map <sup>F</sup> : <sup>→</sup> <sup>G</sup> from (6.3) is given by mapping <sup>I</sup> <sup>∈</sup> <sup>0</sup> to the (uniquely determined) corresponding element g <sup>∈</sup> G in (6.8).

Let π1(M, x, y) denote the set of homotopy classes of paths in M from x to y. We identify the elements in <sup>0</sup> with homotopy classes of paths starting at <sup>x</sup> via the following bijective map:

$$
\pi\_1(M, \mathbf{x}, \mathbf{g}\mathbf{x}) \to F^{-1}(\mathbf{g}) \subset \Gamma\_0,\tag{6.9}
$$

$$
[a] \mapsto I\_a \colon [w] \mapsto [a] \ast [\![g\,w\!] .]$$

It can be easily checked that <sup>I</sup> <sup>−</sup><sup>1</sup> a ([w]) = [g−1a−1]∗[g−1w]. To prove commutativity of the upper square of diagram (6.7), we go around the square both ways.

• *Computing* γ<sup>0</sup><sup>x</sup> (conjγ<sup>0</sup> (γ )). We first describe the conjugation action in terms of concatenation. Using (6.9), we identify γ<sup>0</sup> with a map Ia : <sup>M</sup>- → M for a path <sup>a</sup> satisfying a(0) <sup>=</sup> x and a(1) <sup>=</sup> gx. Similarly, we identify γ with a map Ic : M- → M for a path c satisfying c(0) <sup>=</sup> c(1) <sup>=</sup> x. For any y = [w] ∈ M- with a path w in M starting at w(0) <sup>=</sup> x, we have γ0 (γ )(-

$$\operatorname{conj}\_{\mathbb{M}}(\wp)(\widetilde{\mathbf{y}}) = I\_a(I\_c(I\_a^{-1}([w]))) = [a] \* [\![gc] \* [a^{-1}] \* [w]]$$

and, in particular,

conjγ0 (γ )(x) = [a]∗[gc]∗[a−1].

Now we evaluate γ<sup>0</sup><sup>x</sup> (conjγ<sup>0</sup> (γ )). We first note that

$$\begin{array}{c} \text{Sym}\_{\mathcal{V}} (\mathcal{X} \times \mathcal{X} \times \mathcal{X} \times \cdots \times \mathcal{X}) \\\\ (\text{conj}\_{\mathcal{V}^0} (\mathcal{Y})). \text{ We first note that} \\\\ \mathcal{Y}^0 \widetilde{\mathcal{X}} = I\_a(\widetilde{\mathcal{X}}) = [a] \* (g\widetilde{\mathcal{X}}) = [a], \end{array}$$

since gx is the homotopy class of the constant path at gx in M. This implies

$$\text{monotopy class of the constant path at } g \ge \text{in } M.$$

$$\Phi\_{\mathcal{Y} \widetilde{\text{cr}}}(\text{conj}\_{\mathcal{Y}0}(\mathcal{Y})) = [a^{-1}] \* \Phi\_{\widetilde{\text{x}}}(\text{conj}\_{\mathcal{Y}0}(\mathcal{Y})) \* [a]$$

$$= [a^{-1}] \* \text{conj}\_{\mathcal{Y}0}(\mathcal{Y})(\widetilde{\text{x}}) \* [a]$$

$$= [g \,\text{c}].$$

$$\Phi\_{\widetilde{\text{x}}}(\mathcal{Y}). \text{ We have } \mathbf{g} \cdot \Phi\_{\widetilde{\text{x}}}(\mathcal{Y}) = \mathbf{g} \cdot \boldsymbol{\mathcal{Y}}(\widetilde{\mathbf{x}}) = \mathbf{g} \cdot \mathbf{l}\_{\mathcal{L}}(\widetilde{\mathbf{x}})$$

• *Computing* g ·x (γ ). We have <sup>g</sup> ·x) <sup>=</sup> g ·([c] ∗x) <sup>=</sup> [gc].

Since the results of the two computations agree, the proof is finished.

### **6.3 Realisability of the Wreath Product**

The following result gives an exact topological criterion for realisation of the wreath product, in terms of the **<sup>F</sup>**[G]-module structure of H1(M, **<sup>F</sup>**).

**Proposition 6.3.1** *Suppose that we have a diagram of Riemannian coverings* 

*with* <sup>M</sup><sup>0</sup> *a developable orbifold and* M,M<sup>1</sup> *manifolds. Fix a prime , set* <sup>C</sup> <sup>=</sup> **<sup>Z</sup>**/**<sup>Z</sup>** *and consider the wreath products* G *and* H-<sup>1</sup> *as in Definition* 5.2.1 (*with* <sup>H</sup> <sup>=</sup> <sup>H</sup>1)*. Let* {g1H<sup>1</sup> <sup>=</sup> <sup>H</sup>1, g2H1,...,gnH1} *denote the cosets in* G/H1*. For any* <sup>g</sup> <sup>∈</sup> <sup>G</sup>*, define the permutation* <sup>i</sup> → g(i) *on* {1,...,n} *and the element* hg,i <sup>∈</sup> <sup>H</sup><sup>1</sup> *via*  ggi <sup>=</sup> gg(i)hg,i*.* 

*Define the* **<sup>F</sup>**[G]*-module <sup>N</sup> as <sup>N</sup>* := (Ind<sup>G</sup> <sup>H</sup><sup>1</sup> **<sup>1</sup>**) <sup>⊗</sup>**<sup>Z</sup> <sup>F</sup>** (*cf. Remark* 5.2.2)*. Then the diagram* (6.10) *can be extended to a diagram of the form* (5.6) *if and only if N is a* **<sup>F</sup>**[G]*-quotient module of* H1(M, **<sup>F</sup>**)*.* 

*Proof* We fix the following for the duration of the proof.



In one direction, assume the extended cover exists, corresponding to a quotient of . Now M <sup>→</sup> M is a Galois cover with group the elementary abelian -group Cn. Using the universal property of abelianisation, the corresponding map <sup>→</sup> Cn, whose kernel we denote by , factors through to a map of **<sup>F</sup>**-vector spaces

$$
\varphi \colon \mathcal{A} \twoheadrightarrow \mathcal{C}^n \cong \mathcal{A}'.
$$

We claim that this is a map of **<sup>F</sup>**[G]-modules. For each <sup>g</sup> <sup>∈</sup> <sup>G</sup> consider the corresponding element g <sup>=</sup> (0, g) <sup>∈</sup> G-. Let <sup>γ</sup><sup>0</sup> <sup>∈</sup> <sup>0</sup> denote any lift of g (i.e., g = γ0 ∈ G- <sup>=</sup> 0/ ). By Lemma 6.2.3 applied to both M and M, g acts on and <sup>g</sup> acts on as conjγ<sup>0</sup> . Hence g acts on / <sup>=</sup> C<sup>n</sup> as conjugation by γ0 <sup>=</sup> g. Now in the semidirect product G-, the action of g on c <sup>∈</sup> C<sup>n</sup> is given by conjugation c → gcg<sup>−</sup>1, which corresponds to the action of G on C<sup>n</sup> via , i.e., gives the **<sup>F</sup>**[G]-module structure *<sup>N</sup>* to Cn. Therefore, ϕ(gm) <sup>=</sup> gϕ(m), as desired.

Conversely, suppose there exists a map ϕ : *<sup>M</sup>* - *<sup>N</sup>* of **<sup>F</sup>**[G]-modules. Precomposing with <sup>0</sup> as in (6.2), we get a map : <sup>→</sup> <sup>C</sup>n, and we set

$$
\Gamma' := \ker \Psi \prec \Gamma \text{ and } M' := \Gamma' \backslash M. \tag{6.11}
$$

With this map we can also extend the commutative diagram (6.6) as follows:

where the commutativity of the bottom square is guaranteed precisely by our assumption that <sup>ϕ</sup> is a map of **<sup>F</sup>**[G]-modules.

It remains to verify that the manifold M fits into a diagram (5.6). For this, we need to prove that is normal in <sup>0</sup> (hence also normal in and 1) such that there are induced group isomorphisms 0/ ∼= <sup>G</sup>-, 1/ ∼= H-<sup>1</sup> and / ∼= <sup>C</sup>n, where G- <sup>=</sup> C<sup>n</sup> - G is the wreath product introduced in Definition 5.2.1 (with H <sup>=</sup> H1). For this, we show that the group has the following properties (i)–(v).

(i) *is normal in* 0; indeed, commutativity of diagram (6.12) implies that if γ<sup>0</sup> <sup>∈</sup> <sup>0</sup> with F (γ0) <sup>=</sup> <sup>g</sup> and (γ ) <sup>=</sup> <sup>0</sup> (i.e., γ <sup>∈</sup> ), then (conjγ<sup>0</sup> (γ )) <sup>=</sup> (g)(γ ) <sup>=</sup> <sup>0</sup> (i.e., γ0γ γ −1 <sup>0</sup> <sup>∈</sup> ).

Fix a set-theoretic section <sup>G</sup> <sup>→</sup> <sup>0</sup> : <sup>g</sup> → <sup>g</sup> of the map F, i.e., for any <sup>g</sup> <sup>∈</sup> <sup>G</sup> fix any <sup>g</sup> <sup>∈</sup> <sup>0</sup> such that F (g) <sup>=</sup> <sup>g</sup>. Fix an element <sup>c</sup> <sup>∈</sup> with (c) <sup>=</sup> <sup>e</sup><sup>1</sup> (which exists since is surjective), and for <sup>i</sup> <sup>=</sup> <sup>1</sup>,...,n, let ci := gicg−<sup>1</sup> i <sup>∈</sup> 0. Notice that

$$
\Psi(c\_l) = \Phi(\mathfrak{g}\_l)e\_l = e\_l. \tag{6.13}
$$

Then we have the following properties:


$$
\Psi(\overline{\mathfrak{g}}c\_{l}\overline{\mathfrak{g}}^{-1}) \stackrel{(6.12)}{=} \Phi(\mathfrak{g})\Psi(c\_{l}) \stackrel{(6.13)}{=} \Phi(\mathfrak{g})(e\_{l}) \stackrel{(5.1)}{=} e\_{\mathfrak{g}(l)} \stackrel{(6.13)}{=} \Psi(c\_{\mathfrak{g}(l)}) .
$$

It follows that gcig−1 <sup>=</sup> cg(i) and, since is normal in 0, we can interchange left and right cosets and multiply on the right with g to find the result.

By (iii) and (iv), the cosets of 0/ are given by gc k1 <sup>1</sup> ··· <sup>c</sup> kn n <sup>=</sup> <sup>c</sup> k1 g(1) ··· <sup>c</sup> kn g(n)g with <sup>g</sup> <sup>∈</sup> <sup>G</sup> and <sup>0</sup> <sup>≤</sup> <sup>k</sup>1,...,kn < . Define : <sup>0</sup> <sup>→</sup> <sup>G</sup>- <sup>=</sup> C<sup>n</sup> - G by (c k1

$$
\widehat{\Psi}(c\_1^{k\_1} \cdots c\_n^{k\_n} \overline{\mathfrak{g}} \Gamma') = (k\_1, \dots, k\_n, \mathfrak{g}).
$$

(v) Using the commutativity property in (ii), we have for ki, k j ∈ {0, <sup>1</sup>,...,−1}, and g, g <sup>∈</sup> G, 

 c k1 <sup>1</sup> ··· <sup>c</sup>kn n g c k 1 <sup>1</sup> ··· <sup>c</sup> k n n g = c k1 <sup>1</sup> ··· <sup>c</sup>kn n c k 1 g(1) ··· <sup>c</sup> k n g(n) (g · g ) ,

which follows immediately from (i) and (iv).

(vi) *Via a modification of the section* G <sup>→</sup> 0, g → g, *the map becomes a surjective group homomorphism with kernel*  . This follows immediately from (v) if we can choose the section in such a way that (g · g ) <sup>=</sup> g · g . Consider the short exact sequence

$$\begin{array}{ccccc} 0 \longrightarrow & \Gamma/\Gamma' & \longrightarrow & \Gamma\_0/\Gamma' & \stackrel{\alpha}{\longrightarrow} & \Gamma\_0/\Gamma & \longrightarrow & 1, \\ & \parallel & & \parallel & & \\ C^\pi & & & G & & \end{array} \tag{6.14}$$

with the canonical maps. Note that / ∼= <sup>C</sup><sup>n</sup> is a G-module via the action

$$g \cdot \left( c\_1^{k\_1} \cdot c\_n^{k\_n} \Gamma' \right) = c\_{\mathfrak{g}(1)}^{k\_1} \cdot c\_{\mathfrak{g}(n)}^{k\_n} \Gamma'.$$

Since the order of G is coprime to that of Cn, we have the vanishing of group cohomology:

$$\operatorname{H}^{\operatorname{l}}(G,\Gamma/\Gamma') = \operatorname{H}^{\operatorname{2}}(G,\Gamma/\Gamma') = 0$$

[21, IV 2.3, 3.12, 3.13] and hence (6.14) splits; let j denote a group homomorphism <sup>j</sup> : <sup>G</sup> <sup>→</sup> 0/ such that <sup>α</sup> ◦ <sup>j</sup> <sup>=</sup> idG. Redefine the section <sup>G</sup> <sup>→</sup> <sup>0</sup> in such a way that g <sup>=</sup> j (g). Then we have

$$(\overline{\mathfrak{g}} \cdot \overline{\mathfrak{g}}')\Gamma' = \overline{\mathfrak{g}}\Gamma' \cdot \overline{\mathfrak{g}}'\Gamma' = \mathfrak{z}(\mathfrak{g}) \cdot \mathfrak{z}(\mathfrak{g}') = \mathfrak{z}(\mathfrak{g}\mathfrak{g}') = \overline{\mathfrak{g} \cdot \mathfrak{g}'}\Gamma'$$

and : <sup>0</sup> <sup>→</sup> <sup>G</sup> is a surjective group homomorphism with kernel . It follows that induces an isomorphism 0/ ∼= <sup>G</sup>-. We have already seen the isomorphism / ∼= <sup>C</sup><sup>n</sup> in the proof of (ii). To show that 1/ ∼= <sup>H</sup>-<sup>1</sup> and finish the proof, note that c k1 <sup>1</sup> ··· <sup>c</sup> kn n h with ki ∈ {0,...,−1} and <sup>h</sup> <sup>∈</sup> <sup>H</sup><sup>1</sup> are the cosets of 1/ and that the quotient 1/ is isomorphic to the subgroup H-1 of G under the restriction of the isomorphism induced by . 

### **6.4 Main Result**

We can now prove the main result.

**Theorem 6.4.1** *Suppose* <sup>M</sup><sup>1</sup> *and* <sup>M</sup><sup>2</sup> *are two connected closed oriented smooth Riemannian manifolds such that there is a diagram* 

*of finite covers of a developable Riemannian orbifold* M0*, as in* (1.1) *. Then* 

(i) *The diagram* (1.1) *may be extended to a diagram of finite coverings* 

*as in* (1.2)*, where* M *is a connected closed smooth Riemannian manifold* M *with three Galois covers* 

$$q\_1 \colon M \to M\_1 := H\_1 \backslash M,$$

$$q\_2 \colon M \to M\_2 := H\_2 \backslash M$$

$$q \colon M \to M\_0 := G \backslash M.$$

*Suppose furthermore that there exists a prime number* <sup>≥</sup> <sup>3</sup> *such that*

(∗) (Ind<sup>G</sup> <sup>H</sup><sup>1</sup> **<sup>1</sup>**) <sup>⊗</sup>**<sup>Z</sup> <sup>F</sup>** *is an* **<sup>F</sup>**[G]*-quotient module of* H1(M, **<sup>F</sup>**). *Then* 

(ii) *There exists a diagram of Riemannian coverings* 

*as in* (5.6)*, where* C <sup>=</sup> **<sup>Z</sup>**/**Z***,* G- <sup>=</sup> C<sup>n</sup> - G *is the wreath product where* G *permutes the copies of* <sup>C</sup> *in the same way as it permutes the cosets of* <sup>H</sup><sup>1</sup> *in*  G*, and* Hi <sup>=</sup> <sup>C</sup><sup>n</sup> - Hi *are subgroups of* G *corresponding to the groups* Hi*,*  i <sup>=</sup> <sup>1</sup>, <sup>2</sup>*.*  : H-

(iii) *Consider the linear character* 

$$\Xi \colon \tilde{H}\_{\mathbb{I}} \to \mathbf{C}^\* \colon (k\_1, \dots, k\_n, h\_{\mathbb{I}}) \to e^{2\pi i k\_1}/\ell$$

*with* (k1,...,kn) <sup>∈</sup> <sup>C</sup><sup>n</sup> *and* <sup>h</sup><sup>1</sup> <sup>∈</sup> <sup>H</sup>1*. Then the manifolds* <sup>M</sup><sup>1</sup> *and* M<sup>2</sup> *are equivalent Riemannian covers of* <sup>M</sup><sup>0</sup> *if and only if there exists a linear character* χ : H-<sup>2</sup> <sup>→</sup> <sup>U</sup>(1, **<sup>C</sup>**) *such that the multiplicity of zero in the following two pairs of spectra of twisted Laplacians on* <sup>M</sup><sup>1</sup> *and* <sup>M</sup><sup>2</sup> *coincide:*  IndG-IndG-

 $\left(\begin{array}{c} \text{ $H\_2$ } \rightarrow \text{ $U(1,\text{$ \mathsf{C} $)}$  such that the multiplicative  $\mathsf{op}$ ' $\texttt{zero} in $ \mathsf{m} $}\right)$ 
 $\text{$ f $-spectra}$ 
 $\text{$ L $a$ } \text{ldericans on  $M\_1$  and  $M\_2$  coincide:}$ 

$$\sigma\_{M\_1} (\overline{\Xi} \otimes \text{Res}\_{\widetilde{H}\_1}^{\widetilde{\overline{\mathsf{G}}}} \text{Ind}\_{\widetilde{H}\_1}^{\widetilde{\overline{\mathsf{G}}}} \Xi) \text{ and  $\sigma\_{M\_2} (\overline{\chi} \otimes \text{Res}\_{\widetilde{H}\_2}^{\widetilde{\overline{\mathsf{G}}}} \text{Ind}\_{\widetilde{H}\_1}^{\widetilde{\overline{\mathsf{G}}}} \Xi)$ .$$

*and* 

$$
\sigma\_{M\_1}(\overline{\Xi} \otimes \mathrm{Res}\_{\widetilde{H}\_1}^{\widetilde{G}} \mathrm{Ind}\_{\widetilde{H}\_2}^{\widetilde{G}} \chi) \text{ and } \sigma\_{M\_2}(\overline{\chi} \otimes \mathrm{Res}\_{\widetilde{H}\_2}^{\widetilde{G}} \mathrm{Ind}\_{\widetilde{H}\_2}^{\widetilde{G}} \chi).
$$

$$
\ell | H\_2^{\mathrm{ab}}| \text{ linear characters } \chi \text{ on } \widetilde{H}\_2, \text{ and the dimensions}
$$

*There are* |Hab <sup>2</sup>*, and the dimension of the representations involved is the index* [G : H2]*.* 

*Proof* Part (i) holds by Proposition 2.4.1. Part (ii) is shown in Proposition 6.3.1. Then (iii) holds by Corollary 5.3.1 and Proposition 5.3.2, using that the character  given in (iii), is the one constructed in the proof of Proposition 5.2.3 (cf. formula (5.2)), and is G--solitary on H-1. 

*Remark 6.4.2* Using Proposition 3.10.1, the condition on the multiplicity of zero in the spectrum of the indicated twisted Laplacians in Theorem 6.4.1 may be replaced by an equality of their spectral zeta functions, if we assume in addition that <sup>M</sup><sup>1</sup> and <sup>M</sup><sup>2</sup> are isospectral, i.e., ζM<sup>1</sup>,M<sup>1</sup> <sup>=</sup> ζM<sup>2</sup>,M<sup>2</sup> .

### 6.4 Main Result 55

Condition ( ∗) in the main theorem can be varied, as we will see in the next two chapters. This will also produce a geometric realisation of M and a set of examples where the condition holds.

### **Project**

Describe a theory of (homological) conditions for the realisability of general extensions of finite groups <sup>1</sup> <sup>→</sup> A <sup>→</sup> G <sup>→</sup> H <sup>→</sup> <sup>1</sup> with abelian kernel A, extending a given H-covering; note that a theory of abelian coverings of a given manifold—the case where H is trivial—is encoded almost tautologically in the first homology group, since it is the abelinization of the fundamental group.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 7 Geometric Construction of the Covering Manifold**

In this chapter, we provide a geometric construction of a manifold extending a given Galois cover to a wreath product, using composita and fiber products. For this to be possible, a certain assumption on the homology, previously called ( ∗), needs to be strengthened to a new condition ( ∗∗) (equivalent in most cases). To motivate and use this new condition, we first recall the connection between homology of a quotient and coinvariants. Apart from geometric tools, the construction is also based on the vanishing of certain group cohomology, which is used to prove the existence of certain isometries of manifolds. In the final section, we give a universal property of the wreath product in relation to coverings of manifolds, just like there is such a universal property in the theory of Galois extensions of fields.

### **7.1 From Quotient to Submodule**

If is a prime number coprime to |G|, by Maschke's theorem, any short exact sequence of **F**-[G]-modules splits, and condition ( ∗) from Theorem 6.4.1 is equivalent to

> (∗∗) (Ind<sup>G</sup> <sup>H</sup><sup>1</sup> **1**) ⊗**<sup>Z</sup> F** *is an* **F**-[G]*-submodule of* H1(M, **F**-).

### **7.2 Homology of a Quotient as Coinvariants**

We recall the following tool from invariant theory, see, e.g. [21, II §2]. If R is a commutative ring (for us, R is **Z**, **Q** or **F**-), H a finite group, and *M* is a (left) R[H]-module, its *coinvariants* are defined as the R-module *M*<sup>H</sup> := *M*/I*M* where I is the kernel of the augmentation map R[H] → R : khh → kh. An explicit description is given by

$$I\mathcal{M} = \langle h(\mathbf{x}) - \mathbf{x} \colon h \in H, \mathbf{x} \in \mathcal{A} \rangle\_{\mathcal{H}}$$

(by linearly, it suffices to let x run over a set of generators of *M*). Denote the projection map by

$$
\underline{\mathfrak{t}}\_{R} \colon \mathcal{A} \to \mathcal{A}\_{H} = \mathcal{A} / \mathrm{I} \cdot \mathcal{A} \,. \tag{7.1}
$$

When R is clear from the context, we will leave it out of the notation and simply write t for this map.

This map is particularly easy if *M* = R[H]xi is *free* as an R[H]-module with generators xi; then *M*<sup>H</sup> = Rxi with the obvious map, i.e.,

$$\mathbf{x}\_{R}\colon\bigoplus\_{i}\mathbb{R}[H]\mathbf{x}\_{i}\to\bigoplus\_{i}\mathbb{R}\mathbf{x}\_{i}\colon\sum\_{i}\sum\_{h}k\_{h}h\mathbf{x}\_{i}\mapsto\sum\_{i}\left(\sum\_{h}k\_{h}\right)\mathbf{x}\_{i},\tag{7.2}$$

cf. [21, (2.3)].

One may use "transfer" to prove the following (the case of a free action is also in [21, II.(2.4)]).

**Lemma 7.2.1 ([16, III.2.4])** *If* H *is a finite group of isometries of a closed smooth manifold* M *with quotient map* 

$$q: M \to H \backslash M,$$

*and the order of* H *is coprime to the characteristic of the field* K*, then the first* K*-homology of the quotient,* H1(H\M,K)*, is isomorphic to the coinvariants*  H1(M, K)H *of the first* K*-homology of* M*, and under this identification, the map* q<sup>∗</sup> *that* q *induces on the first homology groups is the map t* <sup>K</sup> *from* (7.1)*, i.e., we have a diagram* 

### **7.3 Geometric Construction**

We refer back to the situation of diagram (6.10), and keep our assumption that **F**- is a field of order coprime to |G|. By condition ( ∗∗), we have a decomposition of **F**-[G]-modules

$$\operatorname{H}\_{\mathcal{l}}(M, \mathbf{F}\_{\ell}) = \mathcal{N} \oplus V \cong \bigoplus \mathbf{F}\_{\ell} \mathcal{O} \oplus V$$

(for some **F**-[G]-submodule V ), where the G-action on *N* is given in terms of the permutation of cosets as gωi = ωg(i), with the convention that i = 1 corresponds to the trivial H1-coset in G. We also let

$$V' := \bigoplus\_{l \ge 2} \mathbf{F}\_{\ell} a\_l$$

denote the vector space complement of **F**ω<sup>1</sup> in *N* .

The quotient map q<sup>1</sup> : M → M<sup>1</sup> = H1\M induces a surjective map

$$q\_{1\*}\colon \operatorname{H}\_{\operatorname{l}}(\operatorname{M}, \operatorname{F}\_{\ell}) \to \operatorname{H}\_{\operatorname{l}}(\operatorname{M}\_{\operatorname{l}}, \operatorname{F}\_{\ell}),$$

and we define ω <sup>1</sup> := q1∗(ω1).

Let <sup>1</sup> denote the subgroup <sup>1</sup> <sup>≤</sup> <sup>0</sup> for which <sup>M</sup><sup>1</sup> <sup>=</sup> 1\M.

**Lemma 7.3.1** *Suppose is coprime to* |G| *and condition* ( ∗) (equivalently, ( ∗∗)) *holds. Then we have a well-defined and commutative diagram:* 

*where* 


$$\varphi\_0 \colon \operatorname{H}\_1(M\_1, \mathbf{F}\_\ell) \xrightarrow{\cong} \mathbf{F}\_\ell \\
\omega\_1' \oplus W \to \mathbf{F}\_\ell \cong \mathbf{C}$$

$$k\_1 \\
\omega\_1' + w \mapsto k\_1 \qquad (k\_1 \in \mathbf{F}\_\ell, \, w \in W) .$$

*with* W := q1∗(V ⊕ V ) *a complementary vector space to* **F**ω <sup>1</sup> *in* H1(M1, **F**-)*.* 

*Proof* To see that this is well defined and the right square commutes, we need that ω <sup>1</sup> is linear independent of W = q1∗(V ⊕ V ); so suppose that there are a1, a<sup>2</sup> ∈ **F**- such that a1ω <sup>1</sup> + a2q1∗(v) = 0 for some v ∈ V ⊕ V . This means that

$$a\_1 a\_1 + a\_2 v \in \ker(q\_{1\*}).\tag{7.4}$$

By Lemma 7.2.1, the kernel of q1<sup>∗</sup> is equal to the kernel of t**F**- , and by definition this kernel is spanned by elements h1(ωi) − ωi (i = 1,...,n) and h1(v) − v for v ∈ V and h<sup>1</sup> ∈ H1. Now


It follows that ker(q1∗) ⊆ V ⊕V , and by (7.4), a1ω<sup>1</sup> ∈ V ⊕V . Since ω<sup>1</sup> is linearly independent from V ⊕ V , we conclude that a<sup>1</sup> = 0, as desired. This guarantees that if ω = kiωi + v ∈ H1(M, **F**-) with v ∈ V (so ϕ(ω) = (k1,...,kn)), then q1∗(ω) = k1ω <sup>1</sup> + w ∈ H1(M1, **F**-) with w ∈ W, so

$$
\varphi\_0(q\_{1\*}(a)) = k\_1 = r\_1(\varphi(a)).
$$

Just like we defined = ker in (6.11), we now set

$$
\Gamma\_1' := \ker \chi\_0 \lhd \Gamma\_1 \text{ and } \mathcal{M}\_1' := \Gamma\_1' \backslash \widetilde{\mathcal{M}} \text{ with covering map } q\_1' \colon \mathcal{M}\_1' \to \mathcal{M}\_1. \tag{7.5}
$$

The following lemma describes the relationship between the group = ker used in Chap. 6, and <sup>1</sup> := ker χ0, the group used in this chapter.

**Lemma 7.3.2** *Suppose is coprime to* |G| *and condition* ( ∗) (equivalently, ( ∗∗)) *holds. The group* = ker *can be expressed in terms of the group*  <sup>1</sup> = ker χ<sup>0</sup> *and a set* {g1,..., gn} *of lifts of* {g1,...,gn} *to* 0*, as* = new*, where* 

$$\Gamma'\_{\text{new}} := \bigcap\_{l=1}^n \overline{\mathfrak{g}}\_l \Gamma'\_1 \overline{\mathfrak{g}}\_l^{-1} \cap \Gamma = \bigcap\_{l=1}^n (\Gamma \cap \overline{\mathfrak{g}}\_l \Gamma'\_1 \overline{\mathfrak{g}}\_l^{-1}) = \bigcap\_{l=1}^n \overline{\mathfrak{g}}\_l (\Gamma \cap \Gamma'\_1) \overline{\mathfrak{g}}\_l^{-1}.\tag{7.6}$$

*Proof* The equalities in (7.6) follow since is normal in 0. It remains to prove new = ker . Notice that it follows from diagram (7.3) that

$$\Gamma\_1' \cap \Gamma = \ker \chi\_0 \cap \Gamma = \{ \gamma \in \Gamma \mid r\_1 \diamond \Psi(\gamma) = 0 \} = \Psi^{-1}(\{0\} \times C^{n-1}).\tag{7.7}$$

Since is surjective, this implies ( <sup>1</sup> <sup>∩</sup> ) = {0} × <sup>C</sup>n−1. Since by definition

$$(\Phi(\mathcal{g}\_i)(\{0\} \times \mathcal{C}^{n-1}) = \mathcal{C}^{i-1} \times \{0\} \times \mathcal{C}^{n-i},$$

from diagram (6.12), we conclude that

$$
\Psi(\overline{\mathbf{g}}\_l(\Gamma\_l' \cap \Gamma)\overline{\mathbf{g}}\_l^{-1}) = \Phi(\mathbf{g}\_l)\Psi(\Gamma\_l' \cap \Gamma) = C^{l-1} \times \{0\} \times C^{n-l},\tag{7.8}
$$

and therefore

$$\Psi(\Gamma\_{\text{new}}') \subseteq \bigcap\_{i} C^{i-1} \times \{0\} \times C^{n-i} = \{0\},$$

so new ⊆ ker .

To prove the reverse inclusion, assume that (γ ) = 0 for some γ ∈ . Then by diagram (6.12) we also have (γ <sup>−</sup><sup>1</sup> <sup>0</sup> γ γ0) = 0 for any γ<sup>0</sup> ∈ 0, so

$$
\gamma\_0^{-1}\gamma\gamma\_0 \in \Psi^{-1}(0) \subseteq \Psi^{-1}(\{0\} \times C^{n-1}) \stackrel{(7.7)}{=} \Gamma\_1' \cap \Gamma.
$$

Therefore γ ∈ γ0( <sup>1</sup> <sup>∩</sup> )γ <sup>−</sup><sup>1</sup> <sup>0</sup> for all γ0, showing that γ ∈ new, so ker ⊆ new. 

*Remark 7.3.3* Standard expressions for the kernel of the restriction and induction of representations (see, e.g., [54, Lemma 5.11]) allow one to give a representationtheoretic description of new. Namely, let χ<sup>0</sup> denote the linear character on <sup>1</sup> given by <sup>χ</sup>0(γ ) <sup>=</sup> <sup>e</sup>2πiχ0(γ )/ where <sup>χ</sup><sup>0</sup> is as in diagram (7.3). Then, with ker <sup>χ</sup><sup>0</sup> <sup>=</sup> ker χ<sup>0</sup> = <sup>1</sup>, we have

$$\ker \mathrm{Res}\_{\Gamma}^{\Gamma\_0} \mathrm{Ind}\_{\Gamma\_1}^{\Gamma\_0} \widetilde{\chi}\_0 = \Gamma \cap \ker \mathrm{Ind}\_{\Gamma\_1}^{\Gamma\_0} \widetilde{\chi}\_0 = \Gamma \cup \bigcap\_{\gamma \in \Gamma\_0} \gamma\_0 \ker(\widetilde{\chi}\_0) \nu\_0^{-1} = \Gamma\_{\mathrm{new}}^{\prime}$$

We now perform the following 2-step geometric construction:

(a) For <sup>g</sup> <sup>∈</sup> <sup>G</sup>, "twist" the cover <sup>q</sup><sup>1</sup> : <sup>M</sup> <sup>→</sup> <sup>M</sup><sup>1</sup> by defining <sup>q</sup><sup>g</sup> <sup>1</sup> : M → M<sup>1</sup> by <sup>x</sup> → <sup>q</sup>1(g−1x), and set

$$M\_g'' := M\_1' \times\_{M\_1, q\_1^s} M;$$

corresponding to the following diagram:

The two different M in the diagram are in fact identical, but the maps to M<sup>1</sup> are different.

(b) Iteratively construct the fiber product

$$M'\_{\rm new} := M''\_{\rm g\_1} \times\_M M''\_{\rm g\_2} \times\_M \cdots \times\_M M''\_{\rm g\_n},\tag{7.10}$$

where {g1,...,gn} is the chosen set of representatives for G/H1; this is presented in the following diagram:

We will prove that this manifold M new is the same as M , the one constructed in the previous chapter.

**Proposition 7.3.4** *Suppose is coprime to* |G| *and condition* ( ∗) (equivalently, ( ∗∗)) *holds.* 

(i) *The fiber product* M new *in* (7.10) *is represented as* 

$$M\_{\text{new}}' = \left\{ (\mathbf{x}\_1, \dots, \mathbf{x}\_n, \mathbf{x}) \in M\_1' \times \dots \times M\_l' \times M \mid \right. $$

$$q\_1'(\mathbf{x}\_l) = q\_1(\mathbf{g}\_l^{-1}\mathbf{x}), \ i = 1, \dots, n \right\}, \qquad (7.12)$$

*and in these coordinates, the projection* M new → M gi *is given by* 

$$M'\_{\text{new}} \ni (\mathfrak{x}\_1, \mathfrak{x}\_2, \dots, \mathfrak{x}\_n, \mathfrak{x}) \mapsto (\mathfrak{x}\_l, \mathfrak{x}) \in M''\_{\text{gl}}.$$

M new *is a connected manifold and corresponds to the subgroup* new*, so that in fact* M new = M *.* 

	- <sup>c</sup> <sup>=</sup> (ci) <sup>∈</sup> <sup>C</sup><sup>n</sup> <sup>≤</sup> <sup>G</sup> *acts componentwise on each factor* <sup>M</sup> gi *, i.e.,*

$$\iota^{-1}\underline{\mathfrak{C}}\iota\cdot(\mathbf{x}\_1,\mathbf{x}\_2,\dots,\mathbf{x}\_n,\mathbf{x}) = (c\_1\mathbf{x}\_1,\dots,c\_n\mathbf{x}\_n,\mathbf{x});\tag{7.13}$$

$$\text{• } g \in G \le \ddot{G} \text{ acts on } M'\_{\text{new}} \text{ by:}$$

$$\boldsymbol{\mu}^{-1}\boldsymbol{g}\boldsymbol{\iota}\cdot(\mathbf{x}\_{1},\mathbf{x}\_{2},\ldots,\mathbf{x}\_{n},\mathbf{x}) = (\mathbf{x}\_{\mathbf{g}^{-1}(1)},\mathbf{x}\_{\mathbf{g}^{-1}(2)},\ldots,\mathbf{x}\_{\mathbf{g}^{-1}(n)},\mathbf{g}\mathbf{x}),\qquad(7.14)$$

*where* <sup>g</sup>−1(i) *is defined, as before, via* <sup>g</sup>−<sup>1</sup>gi <sup>∈</sup> gg−<sup>1</sup>(i)H1*. Colloquially, this means that, up to an isometry, in diagram* (7.11)*,* g *act naturally on the "base" manifold* M*, while the points in the various* M gj *above a given point in* M *are permuted across these different manifolds in the same way as* g−<sup>1</sup> *permutes the cosets* G/H1*.* 

*Proof* Since the group homomorphism χ<sup>0</sup> : <sup>1</sup> → C in diagram (7.3) is surjective, <sup>1</sup> := ker χ<sup>0</sup> - <sup>1</sup> is of index in 1, and q <sup>1</sup> : M <sup>1</sup> → M<sup>1</sup> is a C-Galois cover.

(a) Since M<sup>1</sup> is a manifold, the compositum is described as

$$M\_{\mathcal{g}}^{\prime\prime} = \{ (\mathbf{x}\_{\mathsf{I}}, \mathbf{x}) \in M\_{\mathsf{I}}^{\prime} \times M \colon q\_{\mathsf{I}}^{\prime}(\mathbf{x}\_{\mathsf{I}}) = q\_{\mathsf{I}}(\mathbf{g}^{-\mathsf{I}}\mathbf{x}) \}.$$

Since the degrees of the covers q <sup>1</sup> : M <sup>1</sup> → M<sup>1</sup> and q<sup>1</sup> : M → M<sup>1</sup> are coprime, Lemma 2.3.2 implies that M <sup>g</sup> is connected and equal to the compositum. As in (6.5), the action of g−<sup>1</sup> on M<sup>0</sup> and M<sup>1</sup> corresponds to the action on <sup>0</sup> and the subgroup <sup>1</sup> by conjugation with <sup>g</sup>−1, where <sup>g</sup> is a lift of <sup>g</sup> to 0. Hence the corresponding group is the intersection <sup>g</sup> := ∩ g <sup>1</sup>g−1, i.e., <sup>M</sup> <sup>g</sup> = <sup>g</sup> \M. By Lemma 2.3.3 and coprimality of the degree, the covering M <sup>g</sup> → M is C-Galois.

(b) Since M is a manifold, the underlying set of the fiber product is indeed the set theoretic fiber product in (7.12). We next argue that M new is connected, agrees with the compositum, and indeed corresponds to the group new (and hence ) in (7.6), i.e., M new = M . This will finish the proof of (i). To see the connectedness, we use induction with respect to the number of factors. So suppose we have already proven that M <sup>g</sup><sup>1</sup> ×<sup>M</sup> ...M gN−<sup>1</sup> <sup>→</sup> <sup>M</sup> is a connected CN−1-cover corresponding to the group N −1 i=1 gi( ∩ 1)g−<sup>1</sup> <sup>i</sup> . By Lemma 2.3.2,

the product with the next factor M gN is connected if and only if

$$\Gamma = \langle \bigcap\_{l=1}^{N-1} \overline{\mathbf{g}}\_l(\Gamma \cap \Gamma\_1') \overline{\mathbf{g}}\_l^{-1}, \overline{\mathbf{g}}\_N(\Gamma \cap \Gamma\_1') \overline{\mathbf{g}}\_N^{-1} \rangle. \tag{7.15}$$

To prove this, we notice that is true after applying , using (7.8): the image of left hand side is Cn, and the image of the right hand side is the subgroup of C<sup>n</sup> spanned by {0}N−<sup>1</sup> <sup>×</sup> <sup>C</sup>n−<sup>N</sup> and <sup>C</sup>N−<sup>1</sup> × {0} × <sup>C</sup>n−<sup>N</sup> , which equals the whole of Cn. Hence Eq. (7.15) is true up to ker , and from Lemma 7.3.2, it follows that ker is contained in both the left hand side and the right hand side of the equality, proving that (7.15) holds on the nose.

To prove (ii), note that M new <sup>→</sup> <sup>M</sup> is a <sup>C</sup>n-Galois cover by Lemma 2.3.3, with one copy of C acting componentwise on each factor M gi , and this is the same as the action of C<sup>n</sup> on M . The claim about the action of <sup>g</sup> <sup>∈</sup> <sup>G</sup> <sup>≤</sup> <sup>G</sup> can be proven as follows: the action of <sup>G</sup> on <sup>M</sup> is given by considering <sup>G</sup> as a subgroup of G, and as such it acts by isometries on Mnew = M . We know that, in the geometric representation (7.12) for M new,

$$\mathbf{g}\underline{\mathbf{x}} = \underline{\mathbf{y}} \text{ with } \underline{\mathbf{x}} = (\mathbf{x}\_1, \dots, \mathbf{x}\_n, \mathbf{x}), \underline{\mathbf{y}} = (\mathbf{y}\_1, \dots, \mathbf{y}\_n, \mathbf{y})$$

for some *unique* yi ∈ M <sup>1</sup> and y ∈ M with q <sup>1</sup>(yi) (I) <sup>=</sup> <sup>q</sup>1(g−<sup>1</sup> <sup>i</sup> y). We only need to determine what yi and y are. Since the action of G on M is as given, y (II) <sup>=</sup> gx. Recall also that <sup>g</sup>−<sup>1</sup>gi <sup>=</sup> gg−<sup>1</sup>(i)hg,i for some hg,i <sup>∈</sup> <sup>H</sup>1. In particular, with q<sup>1</sup> : <sup>M</sup> <sup>→</sup> <sup>M</sup><sup>1</sup> the covering with group H1, for <sup>x</sup> <sup>∈</sup> <sup>M</sup> we have q1((g−<sup>1</sup>gi)−1x) (III) <sup>=</sup> <sup>q</sup>1(g−<sup>1</sup> <sup>g</sup>−1(i)x). We collect this information to compute

$$q\_1'(\mathbf{y}\_l) \stackrel{(\mathbf{I})}{=} q\_1(\mathbf{g}\_l^{-1} \mathbf{y}) \stackrel{(\mathbf{II})}{=} q\_1(\mathbf{g}\_l^{-1} \mathbf{g} \mathbf{x}) = q\_1((\mathbf{g}^{-1} \mathbf{g}\_l)^{-1} \mathbf{x}) = q\_1(\mathbf{g}\_{\mathbf{g}^{-1}(l)}^{-1} \mathbf{x}) \stackrel{(\mathbf{III})}{=} q\_1'(\mathbf{x}\_{\mathbf{g}^{-1}(l)}).$$

Since q <sup>1</sup> : M <sup>1</sup> → M<sup>1</sup> is a C-cover, this shows that yi = cixg−1(i) for some c = (ci) <sup>∈</sup> <sup>C</sup>n, that a priori depends on <sup>g</sup> and x, i.e., it is a map

$$
\underline{\circ} \colon G \times M' \to C'' .
$$

Let us first prove that it does not depend x ∈ M . Denote the dependency on x by c(x). Let d(·, ·) denote the distance on a manifold induced from the Riemannian metric. Since C acts properly discontinuously on M <sup>1</sup> there is a δ > 0 such that, for any two elements c, c ∈ C and x ∈ M <sup>1</sup>, if d(cx, c x) < δ, then c = c. If x is at distance ε from x in M , then so is gx from gx , and hence so is ci(x)xg−1(i) from ci(x )x <sup>g</sup>−1(i) for all i. Hence

$$\begin{split}d(\boldsymbol{c}\_{l}(\underline{\mathbf{x}})\boldsymbol{x}\_{\underline{g}^{-1}(l)},\boldsymbol{c}\_{l}(\underline{\mathbf{x}}')\boldsymbol{x}\_{\underline{g}^{-1}(l)}) \\ \leq & d(\boldsymbol{c}\_{l}(\underline{\mathbf{x}})\boldsymbol{x}\_{\underline{g}^{-1}(l)},\boldsymbol{c}\_{l}(\underline{\mathbf{x}'})\boldsymbol{x}\_{\underline{g}^{-1}(l)}') + d(\boldsymbol{c}\_{l}(\underline{\mathbf{x}'})\boldsymbol{x}\_{\underline{g}^{-1}(l)}',\boldsymbol{c}\_{l}(\underline{\mathbf{x}'})\boldsymbol{x}\_{\underline{g}^{-1}(l)}) \\ = & d(\boldsymbol{c}\_{l}(\underline{\mathbf{x}})\boldsymbol{x}\_{\underline{g}^{-1}(l)},\boldsymbol{c}\_{l}(\underline{\mathbf{x}'})\boldsymbol{x}\_{\underline{g}^{-1}(l)}') + d(\boldsymbol{x}\_{\underline{g}^{-1}(l)}',\boldsymbol{x}\_{\underline{g}^{-1}(l)}) \leq 2\varepsilon,\end{split}$$

(the equality in the above formula holds since ci(x ) is an isometry) and thus ci(x) = ci(x ) as soon as x and x are at distance < δ/2. We conclude that c(x) is locally constant in x, and since M is connected, c is actually independent of x. so that we have a map

$$
\underline{c} \colon G \to \mathcal{C}^n. \tag{7.16}
$$

Now denote the dependence on g by c(g). We will prove that this is a cocycle; note that we write the group operation on C<sup>n</sup> multiplicatively. We observe that for two elements g, h ∈ G,

$$\begin{aligned} (c\_l(gh)\mathbf{x}\_{(gh)^{-1}(l)}, gh\mathbf{x}) &= gh\underline{\mathbf{x}} = \mathbf{g}\left(c\_l(h)\mathbf{x}\_{h^{-1}(l)}, h\mathbf{x}\right) \\ &= (c\_l(\mathbf{g})c\_{\mathbf{g}^{-1}(l)}(h)\mathbf{x}\_{h^{-1}\mathbf{g}^{-1}(l)}, gh\mathbf{x}), \end{aligned}$$

so c(gh) <sup>=</sup> c(g)c(h)g, where the action of <sup>g</sup> on <sup>c</sup> <sup>=</sup> (ci) is given by c<sup>g</sup> := (cg−<sup>1</sup>(i)). This shows that the map c in Eq. (7.16) is a cocycle from G to Cn, and the corresponding first group cohomology class lies in H1(G, Cn). Since |G<sup>|</sup> and <sup>|</sup>Cn| = <sup>n</sup> are coprime, the latter cohomology group is zero [21, III.(10.1)], proving that <sup>c</sup> is a coboundary, i.e., there exists <sup>v</sup> <sup>∈</sup> <sup>C</sup><sup>n</sup> (independent of g) such that c(g) <sup>=</sup> <sup>v</sup>−1v<sup>g</sup> <sup>=</sup> (v−<sup>1</sup> <sup>i</sup> vg−<sup>1</sup>(i)). Consider the isometry

$$\iota \colon \mathcal{M}'\_{\text{new}} \to \mathcal{M}'\_{\text{new}} \colon \underline{\mathfrak{x}} = (\mathfrak{x}\_{l}, \mathfrak{x}) \mapsto (v\_{l}^{-1} \mathfrak{x}\_{l}, \mathfrak{x}).$$

Now

$$\iota^{-1}\mathfrak{g}\iota(\underline{\mathbf{x}}) = \iota^{-1}(c\_l(\mathbf{g})v\_{\underline{g}^{-1}(l)}^{-1}\mathbf{x}\_{\underline{g}^{-1}(l)}, \mathbf{g}\mathbf{x}) = \iota^{-1}(v\_l^{-1}\mathbf{x}\_{\underline{g}^{-1}(l)}, \mathbf{g}\mathbf{x}) = (\mathbf{x}\_{\underline{g}^{-1}(l)}, \mathbf{g}\mathbf{x}),$$

as was claimed. Note also that conjugating by ι commutes with the action of Cn, so it does not change that action.

*Remark 7.3.5* The action of G on <sup>M</sup> new ties up with the group theoretical construction from the previous chapter, as follows. The group <sup>G</sup> ∼= 0/ acts naturally on M = \M via

$$(\chi\_0 \Gamma') \cdot (\Gamma' \widetilde{\chi}) = \Gamma' \cdot (\chi\_0 \widetilde{\chi}).\tag{7.17}$$

The explicit identification between M and M new is given by the map

$$M' = \Gamma' \backslash \widetilde{M} \ni \Gamma' \widetilde{\mathfrak{x}} \mapsto (\Gamma\_1' \overline{\mathfrak{g}}\_1^{-1} \widetilde{\mathfrak{x}}, \dots, \Gamma\_1' \overline{\mathfrak{g}}\_n^{-1} \widetilde{\mathfrak{x}}, \Gamma \widetilde{\mathfrak{x}}) =: (\mathfrak{x}\_1, \dots, \mathfrak{x}\_n, \mathfrak{x}) \in M'\_{\text{new}}, \tag{7.18}$$

where {giH1} represent the cosets of H<sup>1</sup> in G and G → <sup>0</sup> : g → g is a section such that we have eG <sup>=</sup> e<sup>0</sup> , <sup>g</sup>−<sup>1</sup> <sup>=</sup> <sup>g</sup>−<sup>1</sup> and g <sup>=</sup> j (g) with the homomorphism <sup>j</sup> : <sup>G</sup> <sup>→</sup> 0/ representing the splitting of (6.14). The action of <sup>G</sup> ∼= 0/ , transferred from M to M new is then

$$(\chi\_0 \Gamma') \cdot (\Gamma\_1' \overline{\mathfrak{g}}\_1^{-1} \widetilde{\mathfrak{x}}, \dots, \Gamma\_1' \overline{\mathfrak{g}}\_n^{-1} \widetilde{\mathfrak{x}}, \Gamma \widetilde{\mathfrak{x}}) = (\Gamma\_1' \overline{\mathfrak{g}}\_1^{-1} \chi\_0 \widetilde{\mathfrak{x}}, \dots, \Gamma\_1' \overline{\mathfrak{g}}\_n^{-1} \chi\_0 \widetilde{\mathfrak{x}}, \Gamma \chi\_0 \widetilde{\mathfrak{x}}).\tag{7.19}$$

Let <sup>c</sup> <sup>∈</sup> be an element satisfying (c) <sup>=</sup> <sup>e</sup><sup>1</sup> and set ci := gicg<sup>−</sup><sup>1</sup> <sup>i</sup> ∈ , as in Sect. 6.3. Utilising the diagrams (6.12) and (7.3), we see that ci ∈ gj 1g−<sup>1</sup> <sup>j</sup> for all j = i. Thus, (7.19) implies

$$(c\_l \Gamma') \cdot (\mathbf{x}\_1, \dots, \mathbf{x}\_n, \mathbf{x}) = (\Gamma\_1' \overline{\mathfrak{g}}\_1^{-1} c\_l \widetilde{\mathbf{x}}, \dots, \Gamma\_1' \overline{\mathfrak{g}}\_n^{-1} c\_l \widetilde{\mathbf{x}}, \Gamma c\_l \widetilde{\mathbf{x}})$$

$$= (\Gamma\_1' \overline{\mathfrak{g}}\_1^{-1} \widetilde{\mathbf{x}}, \dots, \underbrace{\Gamma\_1' c \overline{\mathfrak{g}}\_l^{-1} \widetilde{\mathbf{x}}}\_{l \text{-th entry}}, \Gamma\_1' \overline{\mathfrak{g}}\_n^{-1} \widetilde{\mathbf{x}}, \Gamma \widetilde{\mathbf{x}}). \quad (7.20)$$

Let <sup>g</sup> <sup>∈</sup> <sup>G</sup>; by definition, we have <sup>g</sup>−<sup>1</sup>gi <sup>1</sup> <sup>=</sup> gg−<sup>1</sup>(i)c−ki(g) <sup>1</sup> for some ki(g) modulo -. This implies that

$$\begin{split} (\overline{\operatorname{g}}\Gamma') \cdot (\mathbf{x}\_{1}, \ldots, \mathbf{x}\_{n}, \mathbf{x}) & \stackrel{(7.19)}{=} (\Gamma\_{1}^{\prime}\overline{\operatorname{g}}^{-1}\_{1}\widetilde{\operatorname{g}}\widetilde{\mathbf{x}}, \ldots, \Gamma\_{1}^{\prime}\overline{\operatorname{g}}\_{n}^{-1}\overline{\operatorname{g}}\widetilde{\mathbf{x}}, \Gamma\overline{\operatorname{g}}\widetilde{\mathbf{x}}) \\ & = \quad (\Gamma\_{1}^{\prime}c^{k\_{1}(\mathfrak{g})}\overline{\operatorname{g}}\_{\mathcal{g}^{-1}(\mathfrak{l})}\widetilde{\mathbf{x}}, \ldots, \Gamma\_{1}^{\prime}c^{k\_{n}(\mathfrak{g})}\overline{\operatorname{g}}\_{\mathcal{g}^{-1}(\mathfrak{n})}\widetilde{\mathbf{x}}, \Gamma\overline{\operatorname{g}}\widetilde{\mathbf{x}}) \\ & \stackrel{(7.20)}{=} (c^{k\_{1}(\mathfrak{g})}\_{1} \cdots c^{k\_{n}(\mathfrak{g})}\_{n}\Gamma^{\prime}) \cdot (\mathbf{x}\_{\mathfrak{g}^{-1}(\mathfrak{l})}, \ldots, \mathbf{x}\_{\mathfrak{g}^{-1}(\mathfrak{n})}, \mathfrak{g} \cdot \mathbf{x}). \end{split}$$

Now <sup>g</sup> → (cki(g) <sup>i</sup> ) n <sup>i</sup>=<sup>1</sup> is a cocycle from <sup>G</sup> to <sup>C</sup><sup>n</sup> <sup>=</sup> / , and since H1(G, Cn) <sup>=</sup> 0, there exists (m1,...,mn) with ki(g) = mg−1(i) − mi (modulo -). Using the commutativity of the elements ci , this implies that if we set <sup>c</sup><sup>0</sup> := <sup>n</sup> <sup>i</sup>=<sup>1</sup> <sup>c</sup> mi <sup>i</sup> , then

$$(c\_0 \boldsymbol{y} \, (\boldsymbol{g}) c\_0^{-1}) \cdot (\boldsymbol{x}\_1, \dots, \boldsymbol{x}\_n, \boldsymbol{x}) = (\boldsymbol{x}\_{\boldsymbol{g}^{-1}(1)}, \dots, \boldsymbol{x}\_{\boldsymbol{g}^{-1}(n)}, \boldsymbol{g} \cdot \boldsymbol{x})$$

for all g ∈ G and all (x1,...,xn, x) ∈ M new. This shows that a copy of G in <sup>G</sup> ∼= 0/ , namely c0j (G)c−<sup>1</sup> <sup>0</sup> , acts on M new via permutation of the first n entries. In other words, it is possible to conjugate the subgroup <sup>G</sup> in G to realise the specific action (7.14) on M new.

### **7.4 Universal Property of the Wreath Product**

The appearance of the wreath product in our constructions becomes less of a surprise given the following universal property, showing that the minimal Galois cover that "contains" a G-cover and a C-cover as in our situation arises from this wreath product (the analogous result in the theory of field extensions is well known, compare [37, 13.7]).

**Proposition 7.4.1** *Let* G *and* C *denote finite groups with* C *cyclic of prime order not dividing the order of* G*. Suppose that we are given Riemannian manifolds*  M,M1, M <sup>1</sup> *and a developable Riemannian orbifold* M<sup>0</sup> *such that* M → M<sup>0</sup> *is* G*-Galois with subcover* M<sup>1</sup> → M0*, and* M <sup>1</sup> → M<sup>1</sup> *is* C*-Galois. If* N → M<sup>0</sup> *is a Galois cover of minimal degree admitting Riemannian covers* N → M *and* N → M <sup>1</sup>*, then the Galois group* <sup>G</sup> *of* <sup>N</sup> *over* <sup>M</sup><sup>0</sup> *is the wreath product* <sup>G</sup> := <sup>C</sup><sup>n</sup> - G*, where* n *is the degree of the cover* M<sup>1</sup> → M<sup>0</sup> (*see Figure* (7.21)*.*)

*Proof* Writing the manifolds M0,M,M1, M <sup>1</sup>, N as quotients of he universal cover <sup>M</sup><sup>0</sup> of <sup>M</sup><sup>0</sup> by the respectively group 0, , 1, <sup>1</sup>, N , the defining properties of N imply that it is the normal closure of the compositum of M <sup>1</sup> and M over M0, and hence

$$\Gamma\_N = \bigcap\_{\mathbb{M} \in \Gamma\_0} \mathbb{M}(\Gamma \cup \Gamma\_1') \lambda\_0^{-1}.$$

First of all, for g ∈ G, choose one element g ∈ <sup>0</sup> that maps to g ∈ 0/ ∼= G. We claim that

$$\Gamma\_N = \bigcap\_{l=1}^n \Gamma\_{\mathfrak{K}} \text{ where } \Gamma\_{\mathfrak{K}} := \overline{\mathfrak{g}}\_l(\Gamma \cap \Gamma\_1')\overline{\mathfrak{g}}\_l^{-1},$$

for {gi} a set of coset representatives for H<sup>1</sup> in G. Indeed, for any γ<sup>0</sup> ∈ <sup>0</sup> we can write γ<sup>0</sup> = giγ<sup>1</sup> for some i ∈ {1,...,n} and some γ<sup>1</sup> ∈ 1, since the cosets of H<sup>1</sup> ∼= 1/ in G ∼= 0/ are g1H1,...,gnH<sup>1</sup> and the cosets of <sup>1</sup> in <sup>0</sup> are therefore g11,..., gn1. The statement now follows from the fact that γ<sup>0</sup> ∈ <sup>0</sup> must lie in one of these cosets gi1; since both and <sup>1</sup> are normal in 1, we have

$$\gamma\_0(\Gamma \cap \Gamma\_1')\gamma\_0^{-1} = (\overline{\mathcal{g}}\_l \gamma\_1)(\Gamma \cap \Gamma\_1')(\overline{\mathcal{g}}\_l \gamma\_1)^{-1} = \overline{\mathcal{g}}\_l \gamma\_1(\Gamma \cap \Gamma\_1')\gamma\_1^{-1}\overline{\mathcal{g}}\_l^{-1}.$$

$$= \overline{\mathcal{g}}\_l(\Gamma \cap \Gamma\_1')\gamma\_1\gamma\_1^{-1}\overline{\mathcal{g}}\_l^{-1} = \overline{\mathcal{g}}\_l(\Gamma \cap \Gamma\_1')\overline{\mathcal{g}}\_l^{-1}.$$

Now since is normal in 0, ≥ N , and we find an exact sequence

$$1 \to \Gamma/\Gamma\_N \to \Gamma\_0/\Gamma\_N \to \Gamma\_0/\Gamma \cong G \to 1.$$

The natural map <sup>ϕ</sup> : <sup>→</sup> <sup>n</sup> i=1 / gi has kernel <sup>n</sup> i=1 gi = N . Next, / gi ∼= C since the index is the prime number -. Finally, we claim that ϕ is surjective. For this, it suffices to find for every i an element γi ∈ with

$$\varphi(\boldsymbol{\gamma}\_l) = \boldsymbol{e}\_l = (0, \dots, 0, 1, 0, \dots, 0) \in \boldsymbol{C}^n.$$

Since C is cyclic of prime order, every non-zero element is a generator, and it suffices to choose γi ∈ ( j=i gj ) \ gi . This is possible since the reasoning in the first paragraph of this proof shows that the latter set is non-empty. In the end, we find a sequence

$$1 \to C^n \to \Gamma\_0 / \Gamma\_N \to G \to 1$$

where G acts on C<sup>n</sup> by permuting the factors like it permutes the cosets of H1, and this finishes the proof.

*Remark 7.4.2* In our setup, the universality property says the following: if we search for the "easiest possible" twisted Laplace operator on M1, meaning associated to the Laplace operator on some prime order cyclic cover of M1, we necessarily arrive at a diagram of the form (5.6).

### **Project**

Assuming condition (∗∗), one can now give the following alternative construction of diagram (5.6) used in the main Theorem 6.4.1: perform the above two step construction of M new and *define* an action of G on <sup>M</sup> new using the right hand side of Eqs. (7.13) and (7.14). Prove directly that this manifold satisfies the required properties.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 8 Homological Wideness**

In this short chapter, we introduce a new topological notion: the action of a finite group G on a manifold is called K*-homologically wide* if the first homology group with coefficients in K contains the regular representation of G; this is in some sense complementary to the notion of homological triviality that is well studied in algebraic topology. We study how homological wideness behaves under reduction modulo primes. We also relate homological wideness to the previous conditions ( ∗) and ( ∗∗) for realisability of certain wreath products as covering groups of manifolds.

### **8.1 The Notion of Homological Wideness**

**Definition 8.1.1** Suppose G is a finite group acting (freely or not) on a closed connected (topological) manifold M. Let K denote a field. We say the action of G is K*-homologically wide* if the K*-homology representation* hK = h of G, given by the induced action on the first homology group

$$h\_K \colon G \to \text{Aut}(\mathcal{H}\_\mathbb{I}(M, K)) \tag{8.1}$$

contains the regular representation of G.

Recall that, for a ring R, an R-module M is called *cyclic* if there exists a *cyclic vector* m ∈ M, i.e., a vector such that Rm = M. The regular representation is a cyclic G-module (in this case, *any* element g ∈ G is a cyclic vector, since the vector space span of the orbit G · g spans the entire representation space), hence another way to formulate homological wideness is as follows: *the action of* G *on*  M *is homologically wide if and only if there exists a class* ω ∈ H1(M, K) *such that the orbit* G · ω *spans a vector space of dimension* |G| *inside* H1(M, K). Indeed, if the regular representation is contained in the homology representation, just take a cyclic vector for that subrepresentation. Conversely, if such ω exists, then all g·ω for g ∈ G are linearly independent, and hence span a copy of the regular representation.

**Lemma 8.1.2** *If the action of* G *on* M *is* **F***-homologically wide, then condition*  (∗∗)*, and hence condition* (∗) *for coprime to* |G|*, holds for any subgroup* H1 *in* G*.* 

*Proof* It suffices to show that for any H1 <sup>≤</sup> <sup>G</sup>, Ind<sup>G</sup> <sup>H</sup>1 **1** is a subrepresentation of the regular representation. For that, it suffices to prove that the multiplicity of any irreducible G-representation ρ in Ind<sup>G</sup> <sup>H</sup>1 **1** is less than or equal to dim ρ, the multiplicity of ρ in the regular representation. By Frobenius reciprocity, we compute that the multiplicity of ρ in Ind<sup>G</sup> <sup>H</sup>1 **1** is

$$\langle \rho, \operatorname{Ind}\_{H\_1}^G \mathbf{1} \rangle = \langle \operatorname{Res}\_{H\_1}^G \rho, \mathbf{1} \rangle = \frac{1}{|H\_1|} \sum\_{h\_1 \in H\_1} \operatorname{tr}(\rho(h\_1)) \le \frac{1}{|H\_1|} |H\_1| \dim \rho,$$

since the trace of ρ(h1) is a sum of dim ρ roots of unity (as ρ(h1) is of finite order). 

### **8.2 The Notion of Q-Homological Wideness**

We first relate **Q**-homological wideness (that has a transparent geometric meaning in terms of cycles) to **F**-homological wideness (that is used in the proof of the main result). If the action of G on M is **Q**-homologically wide, there exists a non-torsion homology class ω ∈ H1(M, **Q**) = H1(M,**Z**) ⊗ **Q** such that {gω} is linearly independent over **Q**. Fix an integer N such that ω := Nω ∈ H1(M,**Z**); then {gω } is a set of **Z**-independent non-torsion homology classes for M. These classes will remain linearly independent modulo infinitely many . In fact, we can use representation theory to say more.

**Lemma 8.2.1** *If the action of* G *on* M *is* **Q***-homologically wide, then it is* **F** *homologically wide for all coprime to* |G|*.* 

*Proof* This follows from the basic theory of modular representations in "good" characteristics. In this proof, we write " R–mod" for the category of finitely generated modules over a ring R.

If *M* is a **Q**[G]-module, then it is a **Q**[G]-module (with **Q** the field of -adic numbers). We let K ⊇ **Q** denote a splitting field for all irreducible representations of G; it suffices to assume that K contains all m-th roots of unity where m runs over all orders of elements of G. Let R ⊇ **Z** denote the ring of integers of K and m its maximal ideal with residue field k := R/m.

Fixing any R[G]-lattice *M* in *M*, we have a reduction map modulo m, producing a k[G]-module *M* = *M* ⊗ k = *M* /m*M* . The *decomposition map*

$$d: K\_0(K[G]-\text{mod}) \to K\_0(k[G]-\text{mod}): [\mathcal{A}'] \to [\overline{\mathcal{A}'}]$$

is an isomorphism for coprime to |G| and an effective map (i.e., positive integral combinations map to positive combinations) (see, e.g., [88, §15.5]; another formulation says that if is coprime to |G|, the Brauer character of the reduction of a G-representation modulo m equals the character of the original representation, see, e.g., [54, Thm. 15.8]). By assumption, any irreducible (K-)representation of G occurs as direct summand in H1(M,**Z**) ⊗ K (with multiplicity its dimension), and hence also every k-irreducible representation occurs as direct summand in H1(M,**Z**) ⊗ k (with the same multiplicity). Since the regular representation **Q**[G] is defined over **Q**, we also find the regular representation **F**[G] as direct summand in H1(M,**Z**) ⊗ **F**. 

*Example 8.2.2* For <sup>G</sup> <sup>=</sup> **<sup>Z</sup>**/2**<sup>Z</sup>** = 1 0 3 −1 acting on *<sup>M</sup>* <sup>=</sup> **<sup>Z</sup>**2, (1, <sup>0</sup>) is a cyclic vector over **Q** but not over **F**<sup>3</sup> (so = 3 is excluded by the reasoning before the lemma). The proof of the lemma does not imply that an integral cyclic vector is a cyclic vector modulo ; just that if one exists, then one exists modulo , as long as is coprime to |G|. In the example, (1, 1) is a cyclic vector over both **Q** and **F**3. ♦

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 9 Examples of Homologically Wide Actions**

An action of a finite group *G* on a manifold *M* is homologically wide if the first homology of the manifold contains the regular representation of the group. In this chapter, we study this notion independently of the rest of this monograph. We first study the case where *M* is a surface and *G* acts freely, using the Lefschetz fixed point formula. We then recall a result of Broughton on the homology representation that allows us to deal with the case of a general action on a Riemann surface. After this, we switch to higher dimensional manifolds. We first recall Curtis's theory of the virtual Lefschetz characters, and use results of Cooper and Long to construct examples and counterexamples to homological wideness in all dimensions ≥ 3. Finally, we study locally symmetric spaces. Property (T) implies that only the case of rank 1 is interesting, where we make some remarks on the relation to automorphic representations and Mostow rigidity. Finally, we study an example of using torsion homology: Mednykh's explicit computation of the homology representation of the Seifert-Weber dodecahedral space, which we identify with a known modular representation through Brauer characters.

### **9.1 Surfaces**

In dimension 2, the situation for a fixed-point free action is clear because the homology representation can be computed using the Lefschetz fixed point theorem [48, Theorem 2C.3].

**Proposition 9.1.1** *A fixed-point free action of a non-trivial finite group G on a closed orientable surface M is* **Q***-homologically wide if and only if M is hyperbolic*  (*i.e., has negative Euler characteristic*)*. In particular, the property is independent of G.* 

*Proof* We will compute the character *χh* of the rational homology representation

$$h = h\_{\mathbf{Q}} \colon G \to \mathbf{H}\_{\mathbb{I}}(M, \mathbf{Q}) .$$

First of all, since we assume any *g* = *e* has no fixed points, the map *g* : *M* → *M* has Lefschetz number 0, i.e.,

$$\operatorname{tr}(\mathbf{g}\_\* | \operatorname{H}\_0(M, \mathbf{Q})) - \operatorname{tr}(\mathbf{g}\_\* | \operatorname{H}\_1(M, \mathbf{Q})) + \operatorname{tr}(\mathbf{g}\_\* | \operatorname{H}\_2(M, \mathbf{Q})) = 0.$$

Since the action of *g*<sup>∗</sup> on H0*(M,* **Q***)* ∼= **Q** and H2*(M,* **Q***)* ∼= **Q** is trivial (induced by the action of *g* on the space of connected components of *M*, respectively the 2-cells, i.e., the one-element sets), both outer terms in this expression are 1, and the middle term is *χh(g)* by definition, so we find that *χh(g)* = 2 for *g* = *e*. For *g* = *e*, on the other hand, we get directly from the definition of the character that *χh(e)* = tr*(e*∗| H1*(M,* **Q***))* = dim H1*(M,* **Q***)* = *b*1*(M)* = 2−*χM.* We conclude that

$$\chi\_h(\mathbf{g}) = \begin{cases} 2 & \text{if } \mathbf{g} \neq e, \\ 2 - \chi\_M \text{ if } \mathbf{g} = e. \end{cases}$$

On the other hand, the character of the regular representation is

$$\chi\_{G, \text{reg}} = \begin{cases} 0 & \text{if } g \neq e, \\ |G| \text{ if } g = e. \end{cases}$$

We can match these expressions, and since representations are isomorphic if and only if their characters are equal, we find

$$h = 2 \cdot \mathbf{1}\_G - \frac{\chi\_M}{|G|} \cdot \rho\_{G, \text{reg}},$$

and hence the regular representation of a non-trivial group *G* occurs inside *h* if and only if *χM <* 0.

The above proposition has no (constant) curvature assumption. In constant curvature but with more general actions, we have the following.

**Proposition 9.1.2** *Any action of a finite group G by* (*not necessarily fixed-point free*) *conformal automorphisms on a closed Riemann surface M is* **Q***-homologically wide if χG*\*<sup>M</sup> <* 0*.* 

*Proof* We rely on the computation of the character of *h* in this branched setting by Broughton in [20, Prop. 2(iii)], using in addition (taking into account the holomorphic structure) the Eichler trace formula. Suppose that the *G*-cover is branched above *t* points. For each branch point, choose a lift to the cover and let *Ci* ≤ *G* denote the (cyclic) stabiliser of that lift (the stabilisers of any lift of a given point are conjugate in *G*). Then

$$\begin{aligned} h &= 2 \cdot \mathbf{1} - \chi\_{G\backslash M} \cdot \rho\_{G, \text{reg}} + \sum\_{i=1}^{t} (\rho\_{G, \text{reg}} - \text{Ind}\_{C\_i}^G \mathbf{1}) \\ &= 2 \cdot \mathbf{1} - \chi\_{G\backslash M} \cdot \rho\_{G, \text{reg}} + \sum\_{i=1}^{t} \text{Ind}\_{C\_i}^G (\rho\_{C\_i, \text{reg}} - \mathbf{1}), \end{aligned}$$

where we have used that the induced representation of the regular representation of *Ci* to *G* is the regular representation of *G*. Since **1** occurs in *ρCi,*reg, the representations occurring in the sum are not virtual (i.e., every irreducible representation of *G* occurs in it with non-negative multiplicity) and therefore *h* contains *ρG,*reg as soon as *χG*\*<sup>M</sup> <* 0.

*Remark 9.1.3* In the "non-orbifold quotient" setting of Proposition 9.1.1, by the Riemann-Hurwitz formula, *χM <* 0 if and only if *χG*\*<sup>M</sup> <* 0. This is no longer true in the setting of Proposition 9.1.2, when we only have *χG*\*<sup>M</sup> <* 0 ⇒ *χM <* 0 but not the other way around.

The following is a detailed version of Corollary 1.2.3.

**Corollary 9.1.4** *Let M*1*, M*<sup>2</sup> *be two commensurable non-arithmetic closed Riemann surfaces. Then they admit a diagram* (1.1) *and, assuming the corresponding orbifold M*<sup>0</sup> *satisfies χM*<sup>0</sup> *<* 0*, isometry of M*<sup>1</sup> *and M*<sup>2</sup> *can be checked by computing the multiplicity of zero in at most* 

$$4((\chi\_{M\_1}\chi\_{M\_2}/(\chi\_{M\_0}^{\text{orb}})^2)!)^2$$

*twisted Laplace spectra, where χ*orb *<sup>M</sup>*<sup>0</sup> *is the orbifold Euler characteristic given by* 

$$
\chi\_{M\_0}^{\rm orb} := \chi\_{M^0} - \sum (1 - 1/n\_l), \tag{9.1}
$$

*with ni the order of the stabiliser group at the orbifold points.* 

*Proof* By Proposition 2.5.2, hyperbolic non-arithmetic commensurable closed Riemann surfaces automatically admit a diagram of the form (1.1), and by Proposition 9.1.2, every group action is **Q**-homologically wide since we assume *χM*<sup>0</sup> *<* 0. Therefore, Theorem 1.2.1 applies. To find a prime number coprime to |*G*|, we can always choose  *>* |*G*|, and by Bertrand's postulate [47, Thm. 418], we can find such a prime ≤ 2|*G*|. Hence we can make the bound in Theorem 1.2.1 weaker by 2|*H*ab <sup>2</sup> | ≤ 4|*G*| 2. To express this entirely in terms of the original diagram, we set *di* to be the degree of *Mi* → *M*0. Notice that the compositum *M*<sup>1</sup> •*M*<sup>0</sup> *M*<sup>2</sup> is of degree at most *d*1*d*<sup>2</sup> over *M*0, and the degree of the normal closure of the compositum is of degree at most *(d*1*d*2*)*! over *M*<sup>0</sup> (see (2.3)). Hence |*G*| ≤ *(d*1*d*2*)*!. Now *di* <sup>=</sup> *χMi /χ*orb *<sup>M</sup>*<sup>0</sup> where *χ*orb *<sup>M</sup>*<sup>0</sup> is the orbifold Euler characteristic given by *χM*<sup>0</sup> − *(*1 − 1*/ni)* for *ni* the order of the stabiliser group at the orbifold points [25, 5.1.3]. We find an upper bound of at most

$$4((d\_1d\_2)!)^2 \le 4((\chi\_{M\_1}\chi\_{M\_2}/(\chi\_{M\_0}^{\text{orb}})^2)!)^2$$

for the number of equalities of multiplicities that needs to be checked.

*Remark 9.1.5* If *M*<sup>1</sup> and *M*<sup>2</sup> as in Corollary 9.1.4 are isospectral, by Weyl's law, they have the same volume. Since *di* = vol*(Mi)/*vol*(M*0*)*, we can then assume that *d*<sup>1</sup> = *d*<sup>2</sup> and *χM*<sup>1</sup> = *χM*<sup>2</sup> .

In Chap. 11, one finds some detailed examples of surfaces with less crude bounds on the required number of equalities.

### **9.2 Using the Virtual Lefschetz Character**

The above arguments in dimension 2 are based on very precise information given by fixed point formulæ. These admit a generalisation to a setup as in diagram (1.2) with *M* of arbitrary dimension, where they can sometimes be used to deduce some information about condition *(*∗∗*)* from Sect. 7.1; more specifically, whether Ind*<sup>G</sup> <sup>H</sup>* **1** is a subrepresentation of the homology representation (over **Q**). For this, we use that our manifolds are closed, and thus admit a regular triangulation, which allows us to apply the work of Curtis [32]. Consider the virtual Lefschetz character of *G* given as

$$\Lambda(\mathbf{g}) = -h\mathbf{Q}(\mathbf{g}) + \sum\_{i \neq 1} (-1)^i \operatorname{tr}(\mathbf{g}\_\* | \operatorname{H}\_i(M, \mathbf{Q})).$$

Then by [32, Prop. 1.6], we have

$$
\langle \mathrm{Ind}\_H^G \mathbf{1}, \Lambda \rangle = \chi(M\_\mathcal{l}), \tag{9.2}
$$

the Euler characteristic of *M*1. The formula provides no information for 3 dimensional manifolds, since then *χ (M*1*)* = 0 and = 0. The next proposition provides an example of a result that can be deduced from such methods.

**Proposition 9.2.1** *if <sup>M</sup> is of dimension* 4 *and χ (M*1*)* <sup>≤</sup> <sup>0</sup>*, then* Ind*<sup>G</sup> <sup>H</sup>* **1** *and h***<sup>Q</sup>** *have at least one irreducible representation in common.* 

*Proof* By Poincaré duality, H3*(M,* **Q***)* = Hom*(*H1*(M,* **Q***),* **Q***)* ∼= H1*(M,* **Q***)*, hence

$$
\Lambda = 2 \cdot \mathbf{1} + \mathrm{H}\_2(M, \mathbf{Q}) - 2h\mathbf{q},
$$

and we conclude from (9.2) that

$$2\langle \mathrm{Ind}\_H^G \mathbf{1}, h\_\mathbf{Q} \rangle = 2 + \langle \mathrm{Ind}\_H^G \mathbf{1}, \mathrm{H}\_2(M, \mathbf{Q}) \rangle - \chi(M\_\mathbf{l}) \ge 2 - \chi(M\_\mathbf{l}) > 0.$$

Such results do not suffice to completely verify whether condition *(*∗∗*)* holds under general topological conditions in higher dimension (i.e., only referring to the vector space structure of H1*(M,* **Q***)*, and not to its **Q**[*G*]-module structure), and indeed, in the next few sections we will see examples showing that this is not possible.

# **9.3 Manifolds of Dimension ≥ 3**

In case of 3-manifolds, the picture can vary widely: it is possible to construct a class of closed 3-manifolds with **Q**-homologically wide group actions, but also hyperbolic 3-manifolds with large isometry group for which only the trivial group action is **Q**-homologically wide. This upgrades to similar results in higher dimensions. The results are direct consequences of the work of Cooper and Long [27] for topological manifolds.

**Proposition 9.3.1** *Suppose N is a smooth compact connected* 3*-manifold with a free smooth action by a finite group G, and let γ denote a smooth simple closed curve in N such that the orbit G* · *γ consists of* |*G*| *disjoint smooth simple closed curves. Let* 

*X* = *N* − N *(G* · *γ )*

*denote the open manifold N with an open regular neighbourhood* N *(G* · *γ ) of the G-orbit of γ removed, and let M denote the double of the manifold X* (*i.e., two copies of X glued together along their boundaries*)*. Assume that the embedding ι*: *∂X* → *X of the boundary induces a surjective map on first homology groups ι*<sup>∗</sup> : H1*(∂X,* **Q***)* → H1*(X,* **Q***). Define a Riemannian metric on M as the pullback of any Riemannian metric on the quotient manifold G*\*M; then if M is any (n* − 3*) dimensional closed smooth connected Riemannian manifold with trivial G-action, M* × *M is an n-dimensional closed smooth connected Riemannian manifold with a free isometric G-action that is* **Q***-homologically wide.* 

*Proof* Since [27] concerns topological manifolds, we start by observing that the double of a *smooth* manifold has a smooth structure compatible with the embedding of the original manifold, but composed with a diffeomorphism on one of the copies, see, e.g., [58, VI.5]. Hence *M* is a smooth compact connected manifold on which the finite group *G* acts smoothly (and properly) without fixed points. Therefore, the quotient *G*\*M* is a smooth compact connected manifold, too, and the quotient map *M* → *G*\*M* is a smooth covering map. Choose a Riemannian structure on the quotient *G*\*M* such that it becomes a closed Riemannian manifold, and make *M* into a closed Riemannian manifold by giving it the pullback Riemannian structure. Now *G* acts on *M* by fixed-point free Riemannian isometries.

Cooper and Long [27, Lemma 2.3] have proven that, by the assumption that *ι*<sup>∗</sup> is surjective,

$$\mathbf{H}\_{\rm l}(M, \mathbf{Q}) = \rho\_{G, \rm reg} \oplus (\rho\_{G, \rm reg} - \mathbf{1}),$$

as *G*-modules. Hence, in particular, H1*(M,* **Q***)* contains the regular representation.

Now *G* acts trivially on *M* , so *G* acts by isometries on the cartesian product *M* × *M* . By the Künneth formula, *M* × *M* has first homology group

$$\mathbf{H}\_{\mathbb{I}}(M \times M', \mathbf{Q}) = \mathbf{H}\_{\mathbb{I}}(M, \mathbf{Q}) \oplus \mathbf{H}\_{\mathbb{I}}(M', \mathbf{Q}),$$

so *ρG,*reg is also a subrepresentation of H1*(M* × *M ,* **Q***)*, and the action of *G* on *M* × *M* is **Q**-homologically wide.

In the other direction, Cooper and Long have also shown that through Dehn surgeries, it is possible to "remove" the canonical **Q**[*G*]-modules *ρG,*reg and also *ρG,*reg−**1** from the homology representation to arrive at a rational homology 3 sphere with a *G*-action. Further surgery along an embedded hyperbolic knot allows one to construct such a *hyperbolic* (i.e., constant −1 curvature) manifold [27, Theorem 2.6].

**Proposition 9.3.2** *For any finite non-trivial group G, there exists a hyperbolic rational homology* 3*-sphere M with a free action of G by isometries on M; in particular, the action of G on M is not* **Q***-homologically wide.* 

We conclude that in dimension 3 homological wideness is unrelated to hyperbolicity (in marked contrast to the case of dimension 2).

**Corollary 9.3.3** *For any finite non-trivial group G, and any dimension n* ≥ 3*, there exists an n-dimensional closed connected Riemannian manifold M with a free action of G by isometries on M for which the action of G on M is not* **Q***homologically wide.* 

*Proof* Let *M* be as in Proposition 9.3.2, and let *G* act trivially on the *(n* − 3*)* dimensional sphere *Sn*−3. Setting *<sup>M</sup>* <sup>=</sup> *<sup>M</sup>* <sup>×</sup> *<sup>S</sup>n*−3, by the Künneth formula, we have H1*(M ,* **Q***)* = 0 for *n* = 4 and H1*(M ,* **Q***)* = **Q** for *n* = 4, so it is impossible for non-trivial *G* to act homologically wide on *M* .

*Remark 9.3.4* Bartel and Page have shown that there exists a closed hyperbolic 3 manifold *M* with a free action of any given finite group *G* by isometries on *M* such that additionally, H1*(M,* **Q***)* is any given **Q**[*G*]-module [7].

# **9.4 Locally Symmetric Spaces of Rank ≥ 2**

Let **G** denote a connected semisimple Lie group with trivial center, **K** a maximal compact subgroup of **G**, and  a discrete subgroup of **G** such that \**G** is compact. Consider the locally symmetric Riemannian manifold *M* := \**G***/***K**. If all factors of **G** have real rank ≥ 2, then  has Kazhdan's property (T), and hence H1*(M,* **Q***)* = ab <sup>⊗</sup>**<sup>Z</sup> <sup>Q</sup>** = {0} (see, e.g., [8, Cor. 1.3.6]). This shows the following.

**Proposition 9.4.1** *Only the trivial group can have a* **Q***-homologically wide action on a locally symmetric space of rank* ≥ 2*.* 

### **9.5 Locally Symmetric Spaces of Rank 1**

On the other hand (keeping the notations of Sect. 9.4), if **G** has rank 1, the first Betti number of *M* can be expressed in terms of representation theory via a formula of Matsushima's [67]; more precisely, a sum of multiplicities of specific representation occurring in the representation *R*  of **<sup>G</sup>** by right multiplication on *L*2*(* \**G***)*. If **G** = SO*(n,* 1*)* for *n* ≥ 3, there is a unique representation *J*<sup>1</sup> in that sum and *b*1*(M)* equals the multiplicity of the representation *J*<sup>1</sup> in *R* . Here, *J*<sup>1</sup> is the unique unitary irreducible representation with non-zero Lie algebra cohomology. Except for *n* = 3, *J*<sup>1</sup> is not in the discrete or principal series ([34, Thm. V.5; Rem. V.8; Prop. V.6]; [50, Lemma 4.4] or [14, VII.4.9]). For *n* = 3, *J*<sup>1</sup> is the principal series representation of PSL*(*2*,* **C***)* on *L*2*(***C***)* with Gelfand-Graev-Vilenkin parameters *(*2*,* 0*)*, given explicitly as

$$J\_1\left(\left(\begin{smallmatrix} a \ b \\ c \ d \end{smallmatrix}\right)\right)(f)(z) := (cz+d)^2 f\left(\frac{az+b}{cz+d}\right).$$

**Proposition 9.5.1** *Let <sup>M</sup>* <sup>=</sup> \H*<sup>n</sup> denote a closed hyperbolic n-manifold (<sup>n</sup>* <sup>≥</sup> <sup>3</sup>*), corresponding to a cocompact discrete subgroup in* SO*(n,* 1*). If G is a finite group acting* **Q***-homologically widely on M, then* 

$$|G| \le \langle J\_1, \mathcal{R}\_\Gamma \rangle,$$

*the multiplicity of the representation J*<sup>1</sup> *described above in the* SO*(n,* 1*) representation R given by right multiplication on <sup>L</sup>*2*(* \SO*(n,* <sup>1</sup>*)).* 

### **9.6 Hyperbolic Manifolds; Formulation in Terms of Uniform Lattices**

If *<sup>M</sup>* <sup>=</sup> \H*<sup>n</sup>* is a compact connected hyperbolic manifold of dimension *n* ≥ 3 with finite full isometry group Isom*(M)*, Mostow rigidity implies that Isom*(M)* ∼= Out*( )*, the outer automorphism group of ; indeed, *M* is an Eilenberg-MacLane *K( ,* 1*)*, and hence Out*( )* is isomorphic to the group of homotopy self-equivalences up to free homotopy; but by Mostow rigidity, every homotopy equivalence is homotopic to an isometry [72, Thm. 24.1']. Hence homological wideness of the action of a subgroup *G* → Isom*(M)* on *M* can be formulated in purely group theoretical terms.

**Proposition 9.6.1** *The action of a finite group G of isometries on a compact connected hyperbolic manifold <sup>M</sup>* <sup>=</sup> \H*<sup>n</sup> of dimension <sup>n</sup>* <sup>≥</sup> <sup>3</sup> *is K-homologically wide if and only if the representation* 

$$G \hookrightarrow \text{Out}(\Gamma) \to \text{Aut}(\Gamma^{\text{ab}} \otimes\_\mathbf{Z} K) \cong \text{GL}(b\_1(\Gamma), K)^\perp$$

*contains the regular representation.* 

Recall again that Proposition 9.3.2 gives an example where this representation is trivial for *K* = **Q**. Belolipetsky and Lubotzky [9] have shown that, given any finite group *G*, there exist infinitely many compact connected hyperbolic manifolds with *G* as isometry group.

*Remark 9.6.2* Besson et al. [12, Théorème 9.1] have proven that *homeomorphic*  oriented hyperbolic manifolds of the same dimension *n* ≥ 3 with the same volume are isometric. In our situation, we start from a "correspondence" as in diagram (1.1) and a homeomorphism is not given.

### **9.7 Using Torsion Homology for Homological Wideness**

Instead of using **Q**-homological wideness, one may try to find suitable for which the action of *G* on H1*(M)* is **F**-homologically wide for specific . For example, H1*(M)* might be torsion, so that no non-trivial group acts **Q**-homologically wide, but nevertheless, H1*(M,* **F***)* can contain **F**[*G*]. We content ourselves with commenting on one example.

*Example 9.7.1* Let *M* denote the Seifert–Weber dodecahedral space, a hyperbolic 3-manifold with first Betti number zero; cf. [98]. By Mostow rigidity, *M* is uniquely described by its fundamental group

 $\langle a\_1, \dots, a\_6 | a\_3^{-1} a\_6 a\_4^{-1} a\_5 a\_2, a\_2^{-1} a\_6 a\_3^{-1} a\_4 a\_1, a\_6 a\_2^{-1} a\_3 a\_5 a\_1^{-1}, \dots \rangle$ 
$$a\_2 a\_4 a\_5^{-1} a\_6 a\_1^{-1}, a\_3 a\_4^{-1} a\_6 a\_5^{-1} a\_1, a\_4 a\_2 a\_5 a\_3 a\_1 \rangle.$$

For the following facts, especially the computation of the homology representation, we refer to Mednykh [68]:


$$r = \begin{pmatrix} 4 \ 2 \ 4 \\ 0 \ 0 \ 2 \\ 0 \ 3 \ 0 \end{pmatrix}, \ c = \begin{pmatrix} 0 \ 1 \ 0 \\ 0 \ 0 \ 1 \\ 1 \ 2 \ 3 \end{pmatrix}.$$

The isometry group *G* = *r* ∼= **Z***/*2**Z** of *M* has a cyclic vector *(*1*,* 1*,* 0*)* in H1*(M,* **F***)*, so the action of *G* on *M* is **F**5-homologically wide (but not **Q**homologically wide). Similarly, the isometry group *<sup>G</sup>* <sup>=</sup>*crc*−1*<sup>r</sup>* ∼= **<sup>Z</sup>***/*3**<sup>Z</sup>** of *M* has cyclic vector *(*1*,* 0*,* 0*)*. ♦

We can identify this homology representation on the nose, as follows. Let

$$\operatorname{sgn} \colon S\_{\mathfrak{F}} \to \operatorname{F}\_{\mathfrak{F}}^{\*}$$

denote the linear character given by the sign of a permutation (modulo 5), and let

$$
\psi \colon \mathbb{S}\_{\mathbb{S}} \to \operatorname{GL}(\mathfrak{Z}, \mathbb{F}\_{\mathbb{S}})
$$

denote the 3-dimensional irreducible representation constructed as follows as composition factor of the standard permutation representation of *S*5. The group *S*<sup>5</sup> acts on *<sup>V</sup>* := **<sup>F</sup>**<sup>5</sup> <sup>5</sup> by permuting the standard basis vectors; consider the quotient *W* = *V /L* by the *S*5-invariant line *L* := **F**<sup>5</sup> · *(*1*,* 1*,* 1*,* 1*,* 1*)* spanned by the all-one vector, and consider the *S*5-invariant hyperplane

$$U := \{ [w = (w\_1, \dots, w\_{\mathfrak{S}})] \in W \colon \sum w\_i = 0 \}$$

in *W* (this makes sense since we work modulo 5). Then the natural induced action of *S*<sup>5</sup> on *<sup>U</sup>* ∼= **<sup>F</sup>**<sup>3</sup> <sup>5</sup> is the 3-dimensional faithful mod-5 representation *ψ*, that turns out to be irreducible.

**Proposition 9.7.2** *The mod-*5 *homology representation of the isometry group S*<sup>5</sup> *of the Seifert–Weber dodecahedral space is irreducible, and can be identified with* 

$$
\rho \cong \text{sgn} \otimes \psi. \tag{9.3}
$$

*Proof* Let *ρ* : *S*<sup>5</sup> → GL*(*3*,* **F**5*)* denote the explicit realisation of the representation, as in Example 9.7.1. To see the isomorphism in (9.3), we compute that the values of the Brauer character of *ρ* on the conjugacy classes of elements of order coprime to 5 (given by cycle type) are as in Table 9.1.

This equals the Brauer character of sgn ⊗ *ψ*, so *ρ* has the same semisimplification, but since sgn ⊗ *ψ* is irreducible, it is actually isomorphic to *ρ*, that is then also automatically irreducible.

**Table 9.1** Brauer character of the mod-5 homology representation for the Seifert–Weber dodecahedral space


*Remark 9.7.3* If *M* is a closed hyperbolic 3-manifold with large dim**F** H1*(M,* **F***)*, there are sometimes explicit lower bounds on the volume of *M*, expounded in works of Culler and Shalen. For example, if for some prime , dim**F** H1*(M,* **F***)* ≥ 5 then vol*(M) >* 0*.*35 [31].

*Remark 9.7.4* The scope of the method of using -torsion in H1*(M)* in establishing homological wideness (or the weaker conditions we outlined) is unclear, although some heuristics can be set up by considering the size of the (-)torsion subgroup. Bader et al. [5, Theorem 1.8] have constructed, for any *α* ≥ 0, sequences {*Mm*} of (non-arithmetic) closed hyperbolic rational homology 3-spheres, converging in Benjamini–Schramm topology, for which

$$|\!|\mathbf{H}\_{\rm I}(M\_m)\_{\rm tors}| \sim e^{\alpha \operatorname{vol}(M\_m)}.\tag{9.4}$$

A conjecture of Bergeron and Venkatesh [10, Conjecture 1.3 for SO*(*3*,* 1*)*, and bottom of page 122] states that if *<sup>M</sup>* <sup>=</sup> \H<sup>3</sup> is closed hyperbolic *arithmetic*  manifold and *Mn* :=  *m*\H<sup>3</sup> for a chain of congruence subgroups of  *m* of  with trivial intersection, then the growth in (9.4) holds with *α* = 1*/(*6*π ).*

### **Project**

Explicitly determine the isometry group of rational homology 3-spheres and its representation on the torsion in the first homology. Which (modular) representations can occur? (cf. Remark 9.3.4 for the free part.)

### **Open Problem**

In the situation of Remark 9.7.4, understand not just the size, but also (some of) the decomposition of H1*(Mm,* **F***)* as a **Z**[Out*( m)*]-module. A subproblem is, given a hyperbolic manifold \H*<sup>n</sup>* corresponding to a cocompact discrete group  of isometries of hyperbolic *n*-space H*<sup>n</sup>* (*<sup>n</sup>* <sup>≥</sup> 3), to determine the structure of its first homology ab as a **<sup>Z</sup>**[Out*( )*]-module (compare Sect. 9.6).

### **Project**

Develop other topological criteria on manifolds that imply homological wideness for large classes of group actions.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 10 Homological Wideness, "Class Field Theory" for Covers, and a Number Theoretical Analogue**

In a previous chapter, we have constructed a particular Riemannian covering realising a wreath product. In this chapter, we first return to that example and use class field theory for Riemannian coverings (à la Sunada) to study the behaviour of geodesic in such covers. We then relate, in the general case, homological wideness of a group <sup>G</sup> acting on a manifold M (i.e., the question whether the first homology of M contains the regular representation of G) to the existence of geodesics with certain splitting behaviour. In exact analogy to an classical argument in analytic number theory, we use the Ruelle zeta function to show the existence of infinitely many totally split geodesics for a given covering in the negative curvature case. Finally, the analogy with class field theory allows us to study an analogue of homological wideness in the theory of extensions of number fields.

### **10.1 Abelian Class Field Theory Applied to the Cover**  *M***-→** *M*

We briefly revert to the setup of Chap. 7, now under the stronger assumption that the action of <sup>G</sup> on <sup>M</sup> is **<sup>F</sup>**--homologically wide. In this situation, the intermediate covers M-- gi can be characterised in terms of the splitting behaviour of certain geodesics, as we now explain.

First of all, fix <sup>ω</sup> <sup>∈</sup> H1(M, **<sup>F</sup>**-) to be a cyclic vector for the G-action, so that by assumption

$$\mathcal{H}\_{\mathbb{L}}(M, \mathbf{F}\_{\ell}) = \mathbf{F}\_{\ell}[G] \boldsymbol{\omega} \oplus U$$

for some complementary **<sup>F</sup>**-[G]-module U. Recall also that we have chosen a set of representatives {g1,...,gn} for the left cosets G/H1, so {g−<sup>1</sup> <sup>1</sup> ,...,g−<sup>1</sup> n } is a set of representatives for the right cosets H1\G (by Hall's marriage theorem, there even exists a set {gj } that simultaneously represents the left and right cosets, so alternatively we could use the same set of representatives, if chosen suitably.) This allows us to decompose H1(M, **<sup>F</sup>**-) <sup>=</sup> **<sup>F</sup>**-[G]ω <sup>⊕</sup> U <sup>=</sup> U- <sup>⊕</sup> U as direct sum of **<sup>F</sup>**-[H1]-modules, where U- = n j=<sup>1</sup> **<sup>F</sup>**-[H1]g−<sup>1</sup> j <sup>ω</sup> is, by the assumption of homological wideness, a free **<sup>F</sup>**-[H1]-module with basis ωj := <sup>g</sup>−<sup>1</sup> j <sup>ω</sup>. Thus, the quotient map to the H1-coinvariants on U is as described in (7.2), and since this can be identified with the map <sup>q</sup>1<sup>∗</sup> by Lemma 7.2.1, we find an isomorphism of **<sup>F</sup>**- vector spaces H1(M1, **<sup>F</sup>**-) <sup>=</sup> -**F**ω- j <sup>⊕</sup> UH<sup>1</sup> , where <sup>ω</sup>- j := <sup>q</sup>1∗(ωj ); in particular, the <sup>ω</sup>- j are linearly independent. With these identifications, the map <sup>q</sup>1<sup>∗</sup> is given as in the following diagram.

$$\begin{aligned} \operatorname{H}\_{\mathbb{I}}(M, \mathbf{F}\_{\ell}) &\xrightarrow{q\_{\mathbb{I}\*}} \operatorname{H}\_{\mathbb{I}}(M\_{\mathbb{I}}, \mathbf{F}\_{\ell}) \\ \bigvee\_{i=1}^{n} &\xrightarrow{\bigvee\_{i=1}^{n}} \operatorname{F}\_{\ell}[H\_{\mathbb{I}}] \omega\_{j} \oplus U \xrightarrow{\bigsqcup\_{i=1}^{n}} \bigoplus\_{j=1}^{n} \mathbf{F}\_{\ell} \,\omega'\_{j} \oplus U\_{H\_{\mathbb{I}}} \\ \\ \bigoplus\_{i=1}^{n} &\sum\_{k,h \in H} k\_{j,h} h \alpha\_{j} + \mu \mapsto \sum\_{j=1}^{n} \left( \sum\_{k \in H} k\_{j,h} \right) \alpha'\_{j} + \underline{\mathfrak{h}} \end{aligned}$$

with ki,h <sup>∈</sup> **<sup>F</sup>**-, u <sup>∈</sup> U. Since every Riemannian covering of M is (isomorphic to) a quotient of M by a subgroup of , every *abelian* cover of M is (isomorphic to) a quotient of [, ]\M, and hence Galois groups of abelian covers of M correspond to quotient groups of H1(M) ∼= ab. The coverings i : <sup>M</sup>-- gi <sup>→</sup> <sup>M</sup> are abelian, and they correspond, by construction, to the surjective maps

$$\begin{aligned} \left(\varphi\_{l}\colon \operatorname{H}\_{1}(M)\xrightarrow{\otimes \operatorname{F}\_{\ell}}\operatorname{H}\_{1}(M,\operatorname{F}\_{\ell})\xrightarrow{q\_{1\*}=\underline{t}}\operatorname{H}\_{1}(M\_{1},\operatorname{F}\_{\ell})=\bigoplus\_{j=1}^{n}\operatorname{F}\_{\ell}\omega'\_{j}\oplus U\_{H\_{1}}\to\cdots\xrightarrow{\circ}\operatorname{F}\_{\ell},\\ \sum\_{j=1}^{n}k\_{j}\omega'\_{j}+\mu\mapsto k\_{l}\bmod\ell.\end{aligned}$$

with kj <sup>∈</sup> **<sup>F</sup>**-, u <sup>∈</sup> U. We let {j } denote a set of linearly independent elements of H1(M) that map to {ωj } in H1(M, **<sup>F</sup>**-).

The analogue of abelian class field theory for manifolds (described by Sunada in [93, §5], compare [90, §4]) allows us to distinguish the different covers i, as follows.

We consider geodesics in M (smooth closed curves in M locally of minimal length) as oriented cycles, forgetting the parametrisation. A *prime geodesic* is a geodesic that is not a multiple of another geodesic. Let IM denote the free abelian group generated by the prime geodesics of M, and let IM <sup>→</sup> H1(M) denote the map that associates to a prime geodesic the homology class of the corresponding closed loop. We denote the kernel of this map by I <sup>0</sup> M, the subgroup of elements that are homologous to zero. The map is surjective: choose any lift of an element in H1(M) ∼= ab to , and consider the free homotopy class of free loops in M corresponding to its conjugacy class; by shortening, that free homotopy class contains a closed geodesic (E. Cartan's theorem, Note IV in his "Leçons sur la géométrie des espaces de Riemann"; see, e.g. [35, Ch. 12, Thm. 2.2]; as is written in that reference, the result of Cartan does not require negative curvature), and that geodesic maps to the given element in H1(M). We conclude that there is an isomorphism IM/I <sup>0</sup> M ∼= H1(M).

If <sup>p</sup> is a prime geodesic on M, let (p|i) <sup>∈</sup> **<sup>F</sup>** denote a generator for the (cyclic) stabiliser of any lift of <sup>p</sup> to M-- gi (since the cover i is abelian, this does not depend on the chosen lift: in general, the stabilisers of different lifts are conjugate). By the orbit-stabiliser theorem, the number of prime geodesics in M-- gi above <sup>p</sup> is given by |**F**-|/ (p|i). This number is either - (the prime geodesic "splits", and (p|i) <sup>=</sup> 0) or 1 (the prime geodesic is "inert", and (p|i) <sup>=</sup> 0).

A main result in abelian class field theory for manifolds, [93, Prop. 7], says, since the cover i is abelian, the group homomorphism IM <sup>→</sup> **<sup>F</sup>** given by <sup>p</sup> <sup>→</sup> (p|i) is surjective with kernel I <sup>0</sup> M · i(IM-- gi ), and we have the following commutative diagram

$$\begin{array}{c} 0 \xrightarrow{} \longrightarrow \text{ker}(\varphi\_{i}) \xrightarrow{} \longrightarrow \text{H}\_{1}(M) \xrightarrow{\varphi\_{i}} \text{F}\_{\ell} \xrightarrow{} \xrightarrow{} 0 \\\\ 1 \xrightarrow{} \quad \quad \downarrow \\ 1 \xrightarrow{} \varpi\_{i}(I\_{M\_{\overline{g}\_{i}}}) \xrightarrow{} I\_{M}/I\_{M}^{0} \xrightarrow{(\cdot,\varpi\_{i})} \text{F}\_{\ell} \xrightarrow{} \longrightarrow 0 \end{array}$$

Since ker(ϕi) consists of the H1-orbit of all homology classes spanned by both the classes j with <sup>j</sup> <sup>=</sup> <sup>i</sup>, as well as the classes in the complement U, we deduce from this diagram the following result.

**Proposition 10.1.1** *The prime geodesics of* M *that are inert in the cover*  i : <sup>M</sup>-- gi <sup>→</sup> <sup>M</sup> *are precisely the prime geodesics in the* <sup>H</sup>1*-orbit of all geodesics whose homology class lies in the one-dimensional subspace* i *of* H1(M)*.* 

Since gi represent different conjugacy classes of <sup>H</sup><sup>1</sup> in G, the subspaces i of H1(M) are distinct, and hence so are the covers i : <sup>M</sup>-- gi <sup>→</sup> <sup>M</sup> in the fiber product (7.11).

### **10.2 Homological Wideness and Geodesics**

The question whether the action of G on M is **<sup>Q</sup>**-homologically wide may be approached by splitting it into two separate questions:

(a) Does there exist of a prime closed geodesic on M whose G-orbit consists of |G<sup>|</sup> distinct geodesics?

(b) Do the loops corresponding to these geodesics become homologous in H1(M, **<sup>Q</sup>**)?

We have no general framework to deal with the second question (notice the example of the loop separating the two tori in the connected sum T 2#T 2; it is nontrivial in the fundamental group, but becomes trivial in the first homology group, since it bounds one of the tori). However, concerning the first question, we can say the following.

**Proposition 10.2.1** *Suppose that* <sup>M</sup> *and* <sup>M</sup><sup>0</sup> *are negatively curved closed Riemannian manifolds and* <sup>M</sup> <sup>→</sup> <sup>M</sup><sup>0</sup> *is a* G*-Galois Riemannian covering. Then there exists infinitely many closed prime geodesics in* <sup>M</sup><sup>0</sup> *that lift to* <sup>|</sup>G<sup>|</sup> *distinct closed prime geodesics in* M*, that hence form one* G*-orbit of such geodesics.* 

*Proof* We use the following analytical argument (similar to the analytic argument that split primes in number fields exist). Let

$$Z\_{M\_0}(\mathfrak{s}) := \prod\_{\mathfrak{p}} (1 - e^{-s\ell(\mathfrak{p})})^{-1}$$

where <sup>p</sup> runs over closed prime geodesics <sup>p</sup> in <sup>M</sup><sup>0</sup> of length -(p) (i.e., the Ruelle zeta function for the geodesic flow on M).

Let h > <sup>0</sup> denote the volume entropy of the universal covering of M (which is also the volume entropy of M0). Since M is negatively curved, the geodesic flow is weak mixing Anosov, so it follows that ZM(s) converges for Re(s) > h but has a pole at s <sup>=</sup> h [75, Prop. 9].

Since the covering M - <sup>M</sup><sup>0</sup> is Galois, a prime geodesic <sup>p</sup> splits into r<sup>p</sup> distinct prime geodesics in M, which are all of the same length, say, f<sup>p</sup> times the length of <sup>p</sup>. Then <sup>r</sup>pf<sup>p</sup> = |G<sup>|</sup> (see [93, §5] or [90, §4]).

Suppose, by contradiction, that the set <sup>S</sup> of geodesics of <sup>M</sup><sup>0</sup> that split completely into <sup>|</sup>G<sup>|</sup> distinct prime geodesics in <sup>M</sup> is finite. Then <sup>f</sup><sup>p</sup> <sup>≥</sup> <sup>2</sup> for all <sup>p</sup> <sup>∈</sup>/ <sup>S</sup>. We find, with <sup>P</sup> running over the closed prime geodesics of M, and for real s>h,

$$Z\_M(s) = \prod\_{\mathfrak{P}} \left( 1 - e^{-s\ell(\mathfrak{P})} \right)^{-1} \le \prod\_{\mathfrak{p}} \left( 1 - e^{-sf\_{\mathfrak{P}}\ell(\mathfrak{p})} \right)^{-r\_{\mathfrak{p}}}$$

$$\le \left( Z\_{M\_0}(2s) \right)^{|G|} \prod\_{\mathfrak{p} \in S} \left( 1 + e^{-s\ell(\mathfrak{p})} \right)^{|G|}. \tag{10.1}$$

Now ZM<sup>0</sup> (2s) converges at <sup>s</sup> <sup>=</sup> <sup>h</sup>, and hence also the right hand side of the inequality (10.1) converges at s <sup>=</sup> h; this contradicts the fact that the left hand side has a pole at s <sup>=</sup> h, and hence shows that S is infinite.

*Remark 10.2.2* The result also follows from the more general Riemannian covering version of the Chebotarev density theorem due to Parry and Pollicott [76, Theorem 3] applied to the trivial conjugacy class in G.

Kojima [57, Prop. 2] gave a different proof for the case of closed orientable hyperbolic 3-manifolds admitting a totally geodesic embedding of a Riemann surface of genus ≥ 3, using projective laminations.

### **10.3 An Analogue of Homological Wideness for Number Fields**

Suppose that G acts on M with (orbifold) quotient M0, and denote, as usual, the fundamental group of <sup>M</sup> by and that of <sup>M</sup><sup>0</sup> by 0. Then, as in Remark 6.2.2, G acts by outer automorphisms on , and hence it acts by automorphisms on the abelianisation ab; in this interpretation, the homology representation is given by <sup>h</sup>: <sup>G</sup> <sup>→</sup> Aut(ab <sup>⊗</sup>**<sup>Z</sup> <sup>Q</sup>**). This has the following analogue in number theory: if K/**<sup>Q</sup>** is a finite Galois extension with Galois group G, and GK := Gal(**Q**/K), <sup>G</sup>**<sup>Q</sup>** := Gal(**Q**/**Q**), we have a short exact sequence <sup>1</sup> <sup>→</sup> GK <sup>→</sup> <sup>G</sup>**<sup>Q</sup>** <sup>→</sup> <sup>G</sup> <sup>→</sup> 1, and hence an action of <sup>G</sup> by outer automorphisms on GK, given by conjugation by any lift of an element of G to G**Q**. This induces a group representation

$$\mathbf{h} \colon G \to \operatorname{Aut}(G\_K^{\operatorname{ab}} \otimes\_\mathbf{Z} \mathbf{Q}) \mathtt{d}$$

The analogue of **Q**-homological wideness in this context is the following.

**Proposition 10.3.1** *The representation* **<sup>h</sup>** *contains the regular representation* **<sup>Q</sup>**[G]*.* 

*Proof* There exists a prime number p that is totally split in K/**<sup>Q</sup>** (which follows from Chebotarev's theorem, or easier manipulation with zeta functions much as in the proof of Proposition 10.2.1). Let <sup>p</sup> denote any prime ideal of K above such p. Consider the reciprocity map from class field theory

$$\vartheta: \mathbf{A}\_{K,f}^\* \to G\_K^{\mathrm{ab}}$$

K

from the finite idele group **A**∗ K,f of K, let <sup>π</sup><sup>p</sup> denote any uniformiser in the <sup>p</sup>completion of K, and let

$$F := \vartheta\left( (1, \dots, 1, \pi\_{\mathfrak{p}}, 1, \dots, 1) \right)$$

denote a "Frobenius" of p; then for any g <sup>∈</sup> G,

$$\lg(F) = \vartheta\left( (1, \dots, 1, \pi\_{\mathfrak{g}(\mathfrak{p})}, 1, \dots, 1) \right).$$

By the assumption of total splitting, all g(p) for g running through G and for <sup>p</sup> a fixed prime above the given p are distinct. Since the kernel of ϑ is the closure of the diagonally embedded <sup>K</sup><sup>∗</sup> in **<sup>A</sup>**<sup>∗</sup> K,f , we see that {g(F ): <sup>g</sup> <sup>∈</sup> <sup>G</sup>} are distinct commuting elements of infinite order in <sup>G</sup>ab K , and hence <sup>F</sup> is a cyclic vector for G.

*Remark 10.3.2* Compared to the case of manifolds, in the number theory case, the group Gab K <sup>⊗</sup>**<sup>Z</sup> <sup>Q</sup>** is of infinite rank (and captures all ramified abelian extensions), whereas H1(M, **<sup>Q</sup>**) is always of finite rank (and captures topological abelian covers). For number fields, subproblem (a) as in Remark 10.2 is answered affirmatively by a similar splitting theorem as Proposition 10.2.1 for manifolds; and problem (b) linear combinations of geodesics becoming homologous—does not occur at all, due to the specific nature of the reciprocity map.

### **Open Problem**

Continuing along the lines of Remark 10.3.2, in [29] it was shown that isomorphism of number fields is equivalent to topological conjugacy of associated dynamical systems built from the reciprocity map. The analogous question for manifolds becomes: associate to a manifold M the dynamical system given by the monoid generated by prime geodesics acting on H1(M,**Z**), where a prime geodesic acts by adding the homology class of the corresponding closed loop; then under what conditions is isometry of two manifolds <sup>M</sup><sup>1</sup> and <sup>M</sup><sup>2</sup> equivalent to the topological conjugacy of the corresponding dynamical systems, where one additionally assumes that the identification of the prime geodesics is length-preserving? (In number fields, the analogue of the additional assumption would be that the map of prime ideals preserves the norm map, but it turns out that this is automatic in that case, given the other assumptions.)

### **Open Problem**

Study (in examples; or find a criterion) whether and how distinct geodesics in a G-orbit become homologous.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 11 Examples Concerning the Main Result**

We study whether it is possible to apply twisted Laplace spectra (and, if so, how many are necessary) to deduce isometry of some well-known examples of isospectral manifolds in the literature, due to Schüth (simply connected manifolds), Ikeda (lens spaces), Vignéras/Linowitz and Voight (arithmetic surfaces), Milnor (lattice examples), Doyle and Rossetti (Tetra and Didi), Sunada (based on grouptheoretical examples from Gerst, Gaßmann and Komatsu), Brooks and Tse (surfaces of small genus, Riemann surfaces of small genus), Barden and Kang (surfaces of genus two), and Miatello and Rossetti (flat manifolds isospectral for all twists by linear characters).

### **11.1 Examples Where Theorem 1.2.1 Does not Apply**

We first give some examples where the conditions of Theorem 1.2.1 are not met.

*Example 11.1.1* Schüth constructed isospectral, non-isometric simply connected manifolds *M*<sup>1</sup> and *M*<sup>2</sup> [87]: these are non-isometric manifolds that are indistinguishable by any twisted spectrum on functions, simply because there is nothing to twist by.

From the perspective of our two conditions, this example already violates the first: if a diagram like (1.1) would exist, then by simple connectedness, *M*<sup>1</sup> and *M*<sup>2</sup> would both be isometric to *M*-0, and hence isometric, which they are not.

Sutton [94] and An et al. [3] constructed further examples of isospectral simply connected manifolds by using an extension of Sunada's method valid for continuous groups, rather than finite groups, as we are considering in this book. ♦

*Example 11.1.2* The isospectral compact Riemann surfaces constructed by Vignéras [97] are described by commensurable arithmetic lattices and by [24, Proposition 3] there does not exist a diagram (1.1), so our first condition is violated. The original examples have Euler characteristic −201600 (corrected value from [62]) but Linowitz and Voight have constructed similar examples of Euler characteristic −10 and proven that this is the maximal Euler characteristic that can occur for the class of "unitive" torsion free Fuchsian groups; cf. [62]. ♦

*Example 11.1.3* Ikeda's isospectral non-isometric lens spaces from Example 4.2.1 have *<sup>M</sup>* <sup>=</sup> *<sup>S</sup>*5, and *M*<sup>1</sup> <sup>=</sup> *L(*11; <sup>1</sup>*,* <sup>2</sup>*,* <sup>3</sup>*)* and *M*<sup>2</sup> <sup>=</sup> *L(*11*,* <sup>1</sup>*,* <sup>2</sup>*,* <sup>4</sup>*)*. In this example, *Hi* ∼= **Z***/*11**Z**. The action is not **Q**-homologically wide; actually, the only cyclic cover of *M*<sup>1</sup> is *M* itself. ♦

*Example 11.1.4* We consider Milnor's example [71] where *Mi* = *<sup>i</sup>*\**R**<sup>16</sup> with *-*<sup>1</sup> = *E*8⊕*E*<sup>8</sup> and *-*<sup>2</sup> = *E*16. The group *-*1*, -*<sup>2</sup> = *-*1+*-*<sup>2</sup> is a lattice, and, as Chen [24, §3] observed, there is a diagram of the form (1.2) with *M* := *(-*<sup>1</sup> ∩ *-*<sup>2</sup>*)*\**R**16, *G* = *(-*<sup>1</sup> + *-*2*)/(-*<sup>1</sup> ∩ *-*<sup>2</sup>*)* and *Hi* = *i/(-*<sup>1</sup> ∩ *-*<sup>2</sup>*)*. One easily checks, by writing down explicit lattice bases that *G* = *H*<sup>1</sup> × *H*<sup>2</sup> is the Klein four-group with *Hi* ∼= **Z***/*2**Z**. This can be done, for example, by using the MAGMA [15] code below.

### **MAGMA Program Code for Computing** *G*

```
L0:=LatticeWithBasis(8,
[2,0,0,0,0,0,0,0,
0,-1,1,0,0,0,0,0,
0,0,-1,1,0,0,0,0,
0,0,0,-1,1,0,0,0,
0,0,0,0,-1,1,0,0,
0,0,0,0,0,01,1,0,
1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2]);
L1:=DirectSum(L0,L0);
L2:=LatticeWithBasis(16,
[2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,
1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2]);
L3:=L1 meet L2;
Index(L2,L3);
Index(L1,L3);
G:=(L1+L2)/L3;
G;
```
Since *G* is abelian, the subgroups *H*<sup>1</sup> and *H*<sup>2</sup> cannot be weakly conjugate (since they are not equal as subgroups of *G*). A cover realising the wreath product cannot exist: the group *G* acts non-trivially on the coset space *G/H*<sup>1</sup> = *H*2, so the wreath product *C*<sup>2</sup> - *G* is non-commutative, and hence cannot occur as subgroup of the (abelian) fundamental group of *M*.

Chen also observed that for these *M*1*, M*2, there exists another diagram of the form (1.2) in which the corresponding groups *are* weakly conjugate, see [24, Prop. 1]. In that example, the group *G* contains an element involving a non-trivial translation. One easily computes the corresponding lattices bases (e.g., again using the MAGMA [15] code below) to see that in that case, *Hi* ∼= *(***Z***/*2**Z***)*12.

### **MAGMA Program Code for Computing** *H*<sup>1</sup> **and** *H*<sup>2</sup>

```
Ls:=LatticeWithBasis(4,
[2,0,0,0,
0,2,0,0,
0,0,0,2,
1,1,1,1]);
Lt:=DirectSum(Ls,Ls);
L:=DirectSum(Lt,Lt);
H1:=L1/L;
H2:=L2/L;
H1;
H2;
```
Again, the wreath product cannot be realised for the same reason as above (the corresponding wreath product is non-commutative but the fundamental group of *M* is commutative). ♦

*Example 11.1.5* Doyle and Rossetti [36] constructed two isospectral, non-isometric closed flat 3-manifolds, called "Tetra" and "Didi". These are commensurable, given as quotients of **R**3*/(***Z**<sup>2</sup> <sup>×</sup> <sup>2</sup>**Z***)* by the action of the two non-isomorphic groups of order 4, but there is no diagram (1.1) (if so, they would be strongly isospectral, but they cannot be isospectral on 1-forms, since they have different first Betti numbers). ♦

**11.2 Examples from Sunada's Construction** 

If *M*<sup>1</sup> and *M*<sup>2</sup> are isospectral via the Sunada construction, then a diagram of the form (1.2) exists by default (possibly with orbifold *M*0). We discuss some "small" examples of closed surfaces (where a group *G* is realised by choosing a compact hyperbolic Riemann surface *M*<sup>0</sup> whose genus is larger than or equal to the number of


**Table 11.1** Examples of Sunada triples (with *p* an odd prime number) from [90, Example 1], [6, 19] and [45, Example 4.1]. We indicate the dimension of the representations involved in Theorem 1.2.1, as well as an upper bound on the number of equalities of spectra that need to be checked. In the last line, we choose = 2 and use the bound in Remark 5.3.3

generators of *G*, so that there is a surjection *π*1*(M*0*) G*). These examples satisfy the requirements of Theorem 1.2.1 and illustrate numerically that, whereas our auxiliary construction involves manifolds and groups of relatively large order and negative Euler characteristic, the dimension of the representations and the number of required twists can be rather small (dictated by the degrees of the corresponding coverings *M* → *Mi*).

*Example 11.2.1* Sunada lists three examples in [90, §1, Example 1–3], for which we indicate in the first three rows of Table 11.1 the dimensions of the representations by which one needs to twist, as well as how many equalities of multiplicities of zero suffice in Theorem 1.2.1. The examples are


The computations of the data in the table in these cases is straightforward, except for the last case, named "Komatsu"; here, we need to find a prime coprime to <sup>|</sup>*G*| = *(p*3*)*!. If we observe that all prime divisors of <sup>|</sup>*G*<sup>|</sup> are *< p*<sup>3</sup> and use Bertrands postulate/Chebyshev's theorem that there is a prime between *p*<sup>3</sup> and 2*p*<sup>3</sup> <sup>−</sup> <sup>2</sup> [47, Thm. 418], we can certainly find <sup>≤</sup> <sup>2</sup>*p*<sup>3</sup> <sup>−</sup> 3. We also use that |*H*ab <sup>2</sup> | ≤ *<sup>p</sup>*2; for this, note the commutator identities 

$$
\begin{split}
\left[\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \right] = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \\
\left[\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \right] = \left[\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \right] = \left(\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, 1\right).
\end{split}
$$

which imply that the abelianisation of the group of upper triangular matrices in GL*(*3*,* **F***p)* with all diagonal entries equal to 1 is isomorphic to *(***Z***/p***Z***)*2. ♦

We now discuss in some more detail follow-up examples of Brooks–Tse and Barden–Kang that have the smallest known genus (in variable or constant curvature). The results are summarised in the fifth and sixth row of Table 11.1.

*Example 11.2.2* Brooks and Tse (see [19] and [17]) constructed closed surfaces of Euler characteristic −4 (genus 3) that are isospectral but not isometric for well-chosen metrics whose curvature is not constant. Here, *G* = PSL*(*2*,* 7*)* = *(*P*)*SL*(*3*,* <sup>2</sup>*)* is the unique simple group of order 168 <sup>=</sup> 23 · <sup>3</sup>· 7 [51, 6.14(4) & 6.15], the automorphism group of the Fano plane **P**2*(***F**2*)*, and *H*1*, H*2 are index 7 subgroups given as 3×3 matrices in SL*(*3*,* 2*)* in which the first column, respectively the first row, is *(*1*,* 0*,* 0*)* (stabilisers of a point and a dual hyperplane in **P**2*(***F**2*)*). *χM* = −25 · 3. The smallest possible that can be chosen is <sup>=</sup> 5, and then *G*-

In this case, *M*0 is an orbifold sphere with 3 singular points of order 7, and is of order 57 · <sup>168</sup><sup>≈</sup> <sup>13</sup> · 106 and *χM* = −25 · <sup>3</sup> · 57. The dimension of the required representations in Proposition 5.3.1, on the other hand, is only 7. Recall that *S*<sup>4</sup> has a unique normal subgroup isomorphic to the Klein four-group (generated by products of two two-cycles) and quotient isomorphic to *S*3 ∼= GL*(*2*,* 2*)* [51, 5.2]. Since *Hi* ∼= *(***Z***/*2**<sup>Z</sup>** <sup>×</sup> **<sup>Z</sup>***/*2**Z***)* - GL*(*2*,* 2*)* ∼= *S*4 has commutator subgroup *A*4, the number of spectral equalities that needs to be checked is just 2 · <sup>5</sup>· |*H*ab <sup>2</sup>| = 20 (cf. Proposition 5.3.2). ♦

*Example 11.2.3* In the same reference, Brooks and Tse (see [19] and [17]) used a different representation of the same group *G* and the same subgroups *Hi* to construct isospectral non-isometric compact Riemann surfaces (surfaces of *constant negative curvature* −1) of Euler characteristic −6 (genus 4). Here, *M*0 is an orbifold torus with a single singular point of order 7. Then *χM* = −24 · 32 · 57, but one needs to check only 20 equalities of spectral multiplicities of 7-dimensional representations. ♦

*Example 11.2.4* A similar construction of such surfaces (not of constant curvature) of Euler characteristic −2 (genus 2) by Barden and Kang [6] has the largest currently known Euler characteristic. In their case, *G* has order 96, and *H*1 and *H*2 are of index 12 in *G*, both isomorphic to **Z***/*2**Z** <sup>×</sup> **<sup>Z</sup>***/*4**Z**. We can choose <sup>=</sup> 5 and *G*- of order 512 · <sup>96</sup><sup>≈</sup> <sup>2</sup>· 1010. With *χMi* = −2 for *<sup>i</sup>* <sup>=</sup> <sup>1</sup>*,* 2, we have *χM* = −16, *χM* = −24 · 512, and the dimension of the required representations is 12. In this case, one needs to check 80 equalities of multiplicities of zero in various spectra. ♦

*Example 11.2.5* We consider an example where the order of |*G*| is odd (the situation can be realised using Riemann surfaces of sufficiently high genus, as above). Note that *G* is forcedly solvable, by the Feit–Thompson theorem. Let *p* denote an odd prime number. As in Guralnick [45, Example 4.1], consider the group *G* of order *p*5 given as the semidirect product

$$G = A \times H,\text{ where } A = (\mathbf{Z}/\mathbf{p}^2 \mathbf{Z} \times \mathbf{Z}/\mathbf{p} \mathbf{Z})\text{ and }\mathbf{H} = (\mathbf{Z}/\mathbf{p} \mathbf{Z} \times \mathbf{Z}/\mathbf{p} \mathbf{Z})\text{ }$$

with the action *<sup>a</sup>* <sup>→</sup> *<sup>a</sup><sup>h</sup>* of *<sup>h</sup>* <sup>∈</sup> *<sup>H</sup>* on *<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>* determined by

$$(1,0)^{(1,0)} = (1,1), \\ (0,1)^{(1,0)} = (p,1), \\ (1,0)^{(0,1)} = (p+1,0), \\ (0,1)^{(0,1)} = (0,1).$$

Now *G* has two subgroups *Hi* ∼= **Z***/p***Z** × **Z***/p***Z** that are not weakly conjugate; more precisely, in the above description,

$$H\_1 = H = \langle ((0,0),(1,0)), ((0,0),(1,0)) \rangle \text{ and } H\_2 = \langle ((0,0),(1,0)), ((p,0),(0,1)) \rangle.$$

Using the smallest prime ≥ 3 coprime to |*G*| in the main theorem, we need to check the following number of equalities: (a) if *p* = 3, 90 equalities, using = 5; (b) if *<sup>p</sup>* = 3, 6*p*2 equalities, using <sup>=</sup> 3. In this case, a better result is possible using Pintonello's method from Remark 5.3.3, where one can choose = 2 at the cost of adding extra identities, leading to 4*p*2 <sup>+</sup> 2 equalities to be checked (this number equals 38 *<sup>&</sup>lt;* 90 for *<sup>p</sup>* <sup>=</sup> 3 and is always smaller than 6*p*2). Note that an earlier similar example with |*G*| = *<sup>p</sup>*6*,* <sup>|</sup>*Hi*| = *<sup>p</sup>*2 can be found in [2, Ch. IV, Ex. 2, p. 63]. ♦

### **11.3 Flat Manifolds Isospectral for All Twists by Linear Characters**

*Example 11.3.1* Miatello and Rossetti constructed non-isometric closed flat manifolds *M*1 and *M*2 that admit non-trivial twists but are twisted isospectral on functions and forms for all twists by linear characters (and hence also any representation that decomposes as a direct sum of such) [70, 4.5]. Twisted Laplacians of such linear characters act on sections of flat line bundles.

The manifolds are constructed as follows (our notations differ from [70]). Consider the group of affine transformations **R**<sup>4</sup> - O*(*4*)* of **R**4. Let *τv* denote the translation by *<sup>v</sup>* <sup>∈</sup> **<sup>R</sup>**4, let

$$
\Lambda := \{ \mathbf{r}\_v \colon v \in \mathbf{Z}^4 \} \cong \mathbf{Z}^4,
$$

and denote the standard basis vectors as *e*1*,...,e*4. Consider the two orthogonal matrices

$$A := \text{diag}(1, 1, -1, -1) \text{ and } A' := \text{diag}(1, -1, 1, -1)$$

and the vectors

$$b\_1 = (e\_2 + e\_4)/2, b\_1' = e\_3/2, b\_2 = e\_2/2, b\_2' = e\_1/2.$$

The manifolds are *Mi* = *<sup>i</sup>*\**R**4 with *<sup>i</sup>* = *Aτbi, A τb i ,* . The *<sup>i</sup>* are Bieberbach groups fitting into an exact sequence

$$1 \to \Lambda \cong \mathbf{Z}^4 \to \Gamma\_l \to \langle A, A' \rangle \cong (\mathbf{Z}/2\mathbf{Z})^2 \to 1.$$

Concerning our two conditions, the situation in this example is as follows. If *<sup>A</sup>* <sup>∈</sup> <sup>O</sup>*(*4*)* is of order two and *<sup>b</sup>* <sup>∈</sup> <sup>1</sup>*/*2**Z**4, then *(Aτb)*<sup>2</sup><sup>=</sup> *τb*+*Ab* <sup>∈</sup> , since *<sup>b</sup>* <sup>+</sup> *Ab* <sup>∈</sup> **<sup>Z</sup>**4; hence for any *<sup>γ</sup>* <sup>∈</sup> *<sup>i</sup>*, we have *<sup>γ</sup>* <sup>2</sup><sup>∈</sup> <sup>≤</sup> *-*<sup>1</sup>∩ *-*2. Therefore, with *-*0 := *-*1*, -*<sup>2</sup> we have *-*0*/ <sup>i</sup>* ∼= *(***Z***/*2**Z***)*2, and a diagram such as (1.1) exists with *M*0 := *-*<sup>0</sup>\**R**4. Then we can set *<sup>M</sup>* := *-*\**R**4 with *-* := *-*<sup>1</sup>∩ *-*2 to get a diagram (1.2). From the presentation of *<sup>i</sup>*, one may compute the intersection to be *-* <sup>=</sup> ∼= **<sup>Z</sup>**4, so *<sup>M</sup>* is a 4-torus. Since *<sup>G</sup>* <sup>=</sup> *-*0*/*  is a group of exponent 2 and order 16, we find *<sup>G</sup>* <sup>=</sup> *<sup>H</sup>*1 <sup>×</sup> *<sup>H</sup>*2 ∼= *(***Z***/*2**Z***)*4 with *Hi* <sup>=</sup> *(***Z***/*2**Z***)*2.

We now look at the second condition. Since *G/H*1 ∼= *H*2 with left *G*-action induced by the multiplication in *H*, it follows that Ind*<sup>G</sup> <sup>H</sup>*1 **1** ∼= **Z**[*H*2] is the (4 dimensional) regular representation of *H*2, i.e., the direct sum of the four linear characters of the Klein four-group. Note that, similarly, Ind*<sup>G</sup> <sup>H</sup>*2 **1** ∼= **Z**[*H*1], and hence Ind*<sup>G</sup> <sup>H</sup>*1 **<sup>1</sup>** ∼= Ind*<sup>G</sup> <sup>H</sup>*2 **1***,* meaning that *H*1 and *H*2 are weakly conjugate subgroups of *G*.

To analyse the homology representation, we note that H1*(M)* ∼= ab ∼= **<sup>Z</sup>**4. The group *G* acts on this through conjugation by (outer) automorphisms, i.e., if *Aτb* represents an element of *<sup>G</sup>* (with *<sup>A</sup>* <sup>∈</sup> <sup>O</sup>*(*4*)* of order two, *<sup>b</sup>* <sup>∈</sup> <sup>1</sup>*/*2**Z**2), then it acts by *τ*−*bAτvAτb* = *τAv.* We conclude that the representation of *G* on H1*(M)* factors through the representation of *A, A* ∼= *(***Z***/*2**Z***)*2 in GL*(*4*,***Z***)*. Looking at characters, this is the regular representation of the Klein four-group. We conclude that

$$\mathbf{H}\_1(M) \cong \operatorname{Ind}\_{H\_1}^G \mathbf{1}$$

as *G*-modules.

We find that the two conditions of our main result are satisfied; although the action of *G* is not homologically wide (*G* is of order 16 but the first Betti number of *M* is only 4), the above calculation shows that condition *(*∗*)* does hold. Using = 2 as in Remark 5.3.3, we find that the manifolds *M*1 and *M*2 can be distinguished by 18 equalities of twisted spectra for 4-dimensional representations. These representation are then forcedly not all direct sums of linear characters.

In a different direction, Gordon, Ouyang and Schüth [41] [86] have shown how to distinguish the manifolds in this example using a *non-flat* Hermitian *line*  bundle. ♦

### **11.4 An Example that Does not Arise from Sunada's Construction**

*Example 11.4.1* We consider a finite group *G* with two non-weakly conjugate subgroups *H*1 and *H*2. Let *M*0 denote a compact hyperbolic Riemann surface of genus larger or equal to the number of generators of *G*, and fix a surjective morphism *π*1*(M*0*) G*. This leads to a diagram of the form (1.2), and homological wideness is satisfied by Proposition 9.1.1. For example, set *G* = *S*4 with *H*1 = *(*1234*)* ∼= **Z***/*4**Z** and *H*2 = *(*12*)(*34*),(*13*)(*24*)* ∼= **Z***/*2**Z**×**Z***/*2**Z**. Since the cycle types in *H*<sup>1</sup> and *H*2 are different, they are not weakly conjugate. We choose *M*0 to be a genus two compact Riemann surface and let = 5; now we can distinguish the genus 7 Riemann surfaces *M*1 and *M*2 using 40 equalities of multiplicities of zero in the spectra of Laplacians twisted by 6-dimensional representations. ♦

### **Project**

For the "Komatsu" example, find representations of smaller dimension twisting by which implies isometry.

### **Project**

Study the use of spectra of operators on non-flat line bundles to detect isometry.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 12 Length Spectrum**

In this chapter, we prove a version of the main result using the length spectrum instead of the Laplace spectrum, in case the manifold is negatively curved. We introduce the *L*-series corresponding to a unitary representation of the fundamental group, recall its convergence properties in relation to volume entropy, and its behaviour in finite covers.

### **12.1** *L***-Series**

We assume that *M* is a connected negatively curved oriented closed Riemannian manifold. Then the following properties hold:


Recall some elements of the theory of prime geodesics (for general *M* but an abelian cover, we treated this in Sect. 10.1). We call [*γ* ] *prime* if the associated geodesic is not a multiple of another geodesic (in the sense of oriented cycles). Let : *M* → *M* denote a *G*-cover for a finite group *G*. Notice that *hM* = *hM* by definition. Above each fixed prime geodesic of *M* lie finitely many prime geodesics of *M* , the set of which carries a transitive *G*-action. For a fixed *γ* mapping to *γ* , we let *(*|*γ )* denote any element of *G* that generates the stabiliser of *γ* . All such elements are conjugate in *G* (in the abelian case, the element does not depend on the choice of *γ* , cf. Sect. 10.1).

Let *ρ* : *G* → U*(N ,* **C***)* denote a representation. Since the determinant takes the same value on conjugate matrices, we have a well-defined associated *L-series LM(ρ , s)* :=

$$L\_M(\rho, s) := \prod\_{\{\boldsymbol{\nu}\}} \det(\mathbf{l}\_N - \rho((\varpi | \boldsymbol{\nu}))e^{-s\ell(\boldsymbol{\nu})})^{-1},$$

where 1*<sup>N</sup>* is the identity matrix of size *N* × *N*. If *ρ*<sup>1</sup> and *ρ*<sup>2</sup> are two representations of *G*, then [76, p, 135]

$$L\_M(\rho\_1 \oplus \rho\_2) = L\_M(\rho\_1) L\_M(\rho\_2). \tag{12.1}$$

Choosing *ρ* = **1**, *LM(***1***, s)* is related to analogues of the Selberg zeta function. Standard theory of Dirichlet series (applying Möbius inversion to the coefficients of the logarithmic derivative) implies that knowledge of *LM(***1***, s)* is equivalent to knowledge of the multiset of lengths {*(γ )*}. Parry and Pollicott and Adachi and Sunada have proven:

**Lemma 12.1.1 ([76, Thm. 1 and 2] and [1, Thm. A])** *The function LM(ρ , s) converges absolutely for* Re*(s) > hM and can be analytically continued to an open set D that contains the half-plane* Re*(*s*)* ≥ *hM. For an irreducible representation ρ, L(ρ , s) is holomorphic in D unless ρ* = **1***. Furthermore, LM(***1***, s) has a simple pole at s* = *hM.* 

### **12.2 Main Result for the Length Spectrum**

In this setup, the analogue of Lemma 3.9.1 is the following:

**Lemma 12.2.1** *Let G be a finite group acting by fixed-point free isometries on a negatively curved Riemannian manifold M with quotient M* = *G*\*M . Set h* := *hM* = *hM . If ρ is any unitary representation of G, then the multiplicity ρ ,* **1** *of the trivial representation in the decomposition of ρ into irreducibles equals*  −ord*s*=*hLM(ρ , s), the order of the pole of LM(ρ , s) at s* = *h.* 

*Proof* Let *D* denote the extended region of convergence as in Lemma 12.1.1. Decompose *<sup>ρ</sup>* <sup>=</sup> *N i*=1 *ρi, ρ ρi* as a sum over irreducible representations *ρi*. Then by formula (12.1), we have

 $\rho\rho\_i$  as a sum over  $\text{птечисиве}$ \\

 $\text{e}\\

$ L\_M(\rho, s) = \prod\_{i=1}^N L(\rho\_i, s)^{\langle \rho\_i, \rho \rangle},

and set *ρ*<sup>1</sup> = **1** for convenience. Applying Lemma 12.1.1, we find from this product decomposition that ord*s*=*hLM(ρ , s)* = **1***, ρ* . 

We have the following further two analogues of results previously shown for the Laplace spectra. The analogue of Lemma 3.8.1 for *L*-series is the following.

**Lemma 12.2.2 ([91, Remark 2 After Lemma 1])** *If M* → *M*<sup>1</sup> → *M*<sup>0</sup> *is a tower of finite Riemannian coverings and M* → *M*<sup>0</sup> *is Galois with group G, M* → *M*<sup>1</sup> *with group H, and ρ* : *H* → U*(N ,* **C***) a representation, then* 

$$L\_{M\_0}(\operatorname{Ind}\_H^G \rho, s) = L\_{M\_1}(\rho, s).$$

The analogue of Proposition 4.1.1 for *L*-series is the following.

**Lemma 12.2.3** *Suppose that we have diagram* (1.2) *and set h* := *hM*<sup>1</sup> = *hM*<sup>2</sup> = *hM; then, for two linear characters χ*1 <sup>∈</sup> *<sup>H</sup>*q1 *and χ*2 <sup>∈</sup> *<sup>H</sup>*q2 *, a representation isomorphism* Ind*<sup>G</sup> <sup>H</sup>*<sup>1</sup> *<sup>χ</sup>*<sup>1</sup> ∼= Ind*<sup>G</sup> <sup>H</sup>*<sup>2</sup> *χ*<sup>2</sup> *is equivalent to* 

$$\operatorname{mod}\_{s=h} L\_{M\_l}(\overline{\chi}\_l \otimes \operatorname{Res}\_{H\_l}^G \operatorname{Ind}\_{H\_l}^G \chi\_j, \mathbf{s}) \tag{12.2}$$

*being the same for the pairs (i, j ) given by (*1*,* 1*), (*2*,* 1*) and (*1*,* 2*), (*2*,* 2*).* 

*Proof* The proof is essentially the same as that of Proposition 4.1.1, but now using Lemma 12.2.1 instead of Lemma 3.9.1. 

We can adopt the reasoning in the proof of the main theorem to find:

**Theorem 12.2.4** *Suppose we have a diagram* (1.1) *of negatively curved Riemannian manifolds with (common) volume entropy h, and suppose that the action of G on M in the extended diagram* (1.2) *is homologically wide. Then M*<sup>1</sup> *and M*<sup>2</sup> *are equivalent Riemannian covers of M*<sup>0</sup> *if and only if the pole orders at s* = *h of a finite number of specific L-series of representations on M*<sup>1</sup> *and M*<sup>2</sup> *is equal. Riemannian covers of <sup>M</sup>*<sup>0</sup> *if and only if there exists a linear character <sup>χ</sup>* : *<sup>H</sup>*-

*More specifically, using the notation of Theorem 6.4.1, M*<sup>1</sup> *and M*<sup>2</sup> *are equivalent*  <sup>2</sup> → **C**<sup>∗</sup> *such that*  ord*s*=*hLM*<sup>1</sup> *(* <sup>⊗</sup> Res*G*-Ind*G*-*)* <sup>=</sup> ord*s*=*hLM*<sup>2</sup> *(χ* <sup>⊗</sup> Res*G*-Ind*G*-

$$\begin{aligned} \text{at} \\ \text{ord}\_{s=h} L\_{M\_1}(\overline{\Xi} \otimes \text{Res}\_{\widetilde{H}\_1}^{\widetilde{G}} \text{Ind}\_{\widetilde{H}\_1}^{\widetilde{G}} \Xi) = \text{ord}\_{s=h} L\_{M\_2}(\overline{\chi} \otimes \text{Res}\_{\widetilde{H}\_2}^{\widetilde{G}} \text{Ind}\_{\widetilde{H}\_1}^{\widetilde{G}} \Xi) \end{aligned}$$

*and* 

$$\operatorname{ord}\_{s=h} L\_{M\_1}(\overline{\Xi} \otimes \operatorname{Res}\_{\widetilde{H}\_1}^{\widetilde{G}} \operatorname{Ind}\_{\widetilde{H}\_2}^{\widetilde{G}} \chi) = \operatorname{ord}\_{s=h} L\_{M\_2}(\overline{\chi} \otimes \operatorname{Res}\_{\widetilde{H}\_2}^{\widetilde{G}} \operatorname{Ind}\_{\widetilde{H}\_2}^{\widetilde{G}} \chi).$$
 
$$\text{are } \ell | H\_2^{\operatorname{ab}} | \text{ linear characters } \chi \text{ on } \widetilde{H}\_2 \text{, and the dimension of the repn}$$

*There are* |*H*ab 2*, and the dimension of the representations involved is the index* [*G* : *H*2]*.* 

*Remark 12.2.5* In case of a surface of constant curvature −1, Theorems 1.2.1 and 12.2.4 are equivalent using a twisted version of the Selberg trace formula as in [49, III.4.10].

### **Open Problem**

In [28, Theorem 3.1], it was proven that two global function fields (corresponding to smooth projective algebraic curves over finite fields) are isomorphic if and only if there is a group isomorphism between their abelianised absolute Galois groups such that the corresponding *L*-series are equal. In [13], it is shown that one may restrict to unramified characters by admitting extensions of the ground field.

For negatively curved (e.g., hyperbolic) manifolds *M*<sup>1</sup> and *M*2, the corresponding question is whether they are isometric if and only if there is an isomorphism *ψ* : H1*(M*1*,***Z***)* ∼= H1*(M*2*,***Z***)* such that

$$L\_{M\_2}(\chi, s) = L\_{M\_1}(\chi \circ \psi, s).$$

for all *χ*∈ Hom*(π*1*(M*2*),***C**∗*)*=Hom*(*H1*(M*2*,***Z***),***C**∗*).*

### **Open Problem**

The geodesic length function defines the *marked length spectrum* [*γ* ] → *(γ )* as a function from conjugacy classes in *π*1*(M)* to **R***>*0. Croke and Otal [30] [74] showed that this characterises the isometry class of *M* in dimension two (in arbitrary dimension, this is an open conjecture of Burns and Katok, see [22, Problem 3.1]; a local version was proven in [44]). This motivates the question: what is the relation between the marked length spectrum and the information encoded in the *L*-series *LM(ρ , s)* where *ρ* runs over all unitary representations of *π*1*(M)* (this information might be called the "twisted length spectrum")?

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **References**


G. Cornelissen, N. Peyerimhoff, *Twisted Isospectrality, Homological Wideness, and Isometry*, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-27704-7


# **Index**

### **B**

Barden, D., 95 Bartel, A., 78 Brooks, R., 95

### **C**

Class field theory (for manifolds), 85, 87 Coinvariants, 57 – relation to homology of quotient, 58 Commensurable manifolds, 14 – nonarithmetic lattices, 14 Compositum (of manifolds), 11 Condition ( ∗∗), 57 Condition ( ∗), 53 – relation to homological wideness, 70 Cooper, D., 77 Coset permutation representation, 22 – as induced representation, 22 – isomorphism as *G*-sets, 23 – isomorphism as representations, 22

### **D**

De Smit, B., 39 Developable orbifold, 9 Doyle, P.G., 93

### **E**

Entropy, 88, 99

### **F**

Fiber product (of orbifolds), 11


First homology group


### **G**

Gaßmann, 4, 94 Gaßmann equivalence, 23 Geodesics – existence of totally split geodesics, 88 G-set, 22 – transitive, 22

### **H**

Homological wideness


© The Author(s) 2023 G. Cornelissen, N. Peyerimhoff, *Twisted Isospectrality, Homological Wideness, and Isometry*, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-27704-7

Homological wideness (*cont.*) – for locally symmetric space in rank 1, 79 – for locally symmetric space in rank ≥ 2, 78 – relation to condition ( ∗) and ( ∗∗), 70 – relation to geodesics, 87 – for surfaces, 73

Homology representation, 69

### **I**

Ikeda, A., 33 Induced representation, 21

### **K**

Kang, H., 95 Komatsu, 94

### **L**

Lefschetz number, 74 Lens spaces, 33, 92 Linowitz, B., 92 Long, D.D., 77 L-series (for a manifold and a representation of the fundamental group), 100 – convergence, 100 – relation with induced representation for finite covering, 101

### **M**

Mackey's theorem, 34 Mednykh, A.D., 80 Miatello, R.J., 96 Milnor, J.W., 92 Monomial structure (on representation), 37 – isomorphism, 37, 38 – standard monomial structure on induced representation, 38

### **N**

Normal closure (of covering of manifolds), 13

### **O**

Orbifold Euler characteristic, 75

### **P**

Page, A., 78 Pesce, H., 33 Pintonello, M., 40 Prime geodesic, 86, 99 – relation to first homology group, 87

### **R**

Regular representation, 21 Restriction of representation, 21 Rossetti, J.P, 93, 96

### **S**

Schüth, 91 Seifert–Weber dodecahedral space, 80 – identification of homology representation, 81 Solitary character, 38 – existence for order ≥ 3, 39 – nonexistence for order 2, 40 – relation to conjugacy of subgroups, 38 Spectral zeta function (of differential operator), 18 – determines spectrum in odd dimension, 19 – does not determine spectrum in even dimension, 19 – relation with spectrum for twisted Laplacians, 27 Spectrum (of differential operator), 18 Strong isospectrality, 33 – spectral characterisation, 34 Sunada, 4, 25, 33, 94

### **T**

Tse, R., 95 Twisted Laplacian, 23 – associated to a finite cover of manifolds, 24 – associated to a representation of the fundamental group, 23 – decomposition of spectrum, 26 – dimension of kernel expressed using spectral zeta functions, 27

– multiplicity of zero in the spectrum and of trivial representation, 27

– spectrum and induced representation, 24

### **V**

Vignéras M.-F., 15, 91 Virtual Lefschetz character, 76 Voight, J., 92

Index 111

### **W**

	- isospectrality does not imply, 33
	- spectral characterisation, 34

Wreath product, 38

