数理解析研究所講究録
第 1571 巻 2007 年 59-81

59

Arithmetic lattices and weak spectral geometry

Arithmetic lattices and weak spectral geometry
D. B. McReynolds

August 23, 207
Abstract

This note is an expansion of three lectures given at the workshop Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces held at Kyoto
University in December of 2006 and wiil appear in the proceedings for thisworkshop.

Introduction
Our attention in this note will be on the non-exceptional real rank one symmetricspaces arising ffom the simple Lie groups $SO(n, 1),$ $SU(n, 1)$ , and $Sp(n, 1)$ and finite volume quotients of these spaces. These spaces and their quotients are known
as real, complex, and quatemionic hyperbolic n-space and real, complex, andquatemionic hyperbolic n-manifolds, respectively. For these spaces, our aim is
2-fold:$(\theta)$

Provide a description of some of the anthmetic quotients of these symmebic

spaces.

(B) Produce interesting examples of closed quotients of these symmetric spaces
with regard to various spectral problems.

These two goals are essentially independent, although in general the forner is the
only means we have for producing examples in general; in particular, to achieve thelatter we are forced to consider arithmetic constructions. We shall take a leisurely
and loose approach to these goals, providing some background but largely leaving
assertions unproven. The reader interested in more detail and rigor is directed to
the references provided below.

60
Arithmetic lattices and weak spectral geometry

Organization of the article This note is organized into five sections. In the firstsection, we briefly recall the definitions of real, complex, and quaternionic hyperbolic -space. In the second section, we provide a description for constructing
certain arithmetic lattices in the associated isometry groups for these spaces. In thethird section, we discuss some recent results on isospectral manifolds modelled onthese symmetric spaces (and more general symmetric spaces of noncompact type).
In the four section, we discuss some recent work on weaker spectral constructions.
In the fifth section, we discuss some variants of Sunada’s method used to produce
the asserted examples Rom Section 4.
$n$

Acknowledgements I gratefully acknowledge the workshop organizer Michihiko Fujii for the invitation to speak and attend the workshop and its success.
I also wish to acknowledge my gratitude to Yoshinobu Kamishima (and Tokyo
Metropolitan University) for handling the logistics of the trip, for several conversations on the topics of this note, and for his kindness during the duration of my stayin Kyoto and Tokyo. In addition, I want to thank Sadayoshi Kojima and Kenneth
Shackleton for their hospitality while in Tokyo and for the invitation to speak at theTokyo Institute of Technology. Much of what I have said on simple length sets andspectra for surfaces came out during several conversations with Chris Leininger;
I also want to thank Greg McShane and Hugo Parlier for conversations on this
topic. It goes almost without saying that my collaborators Chris Leininger, Walter
Neumam, and Alan Reid have extensively contributed to my discussion of weak
spectral equivalences. Indeed, one should consider those sections as writtenjointlywith them though any mistakes are entirely my doing. Finally, I want to express
my deepest appreciation to the workshop attendees for their interest in my lecturesand for numerous simulating conversations. It was truly a pleasure to speak at andattend this workshop and humbling to be in the company of so many wonderfully
gracious and talented mathematicians.

1 Hyperbolic spaces
For completeness, a short section introducing real, complex, and quatemionic hyperbolic space, their isometry groups, and their orbifold quotients is provided below. The reader should look to [48], [14], and [21] for more thorough $\alpha eaunents$of this material.

61

Arithmetic lattices and weak spectral geometry
Notation Throughout,$X$will denote either $R,C$ , or$\mathbb{H}$

. On

$X$

, we have the invo-$lution*defined$ by$x^{*}=\{\begin{array}{ll}identity, X=Rcomplex conjugation, X=Cquatemionic conjugation, X=\mathbb{H}.\end{array}$

We extend this to a map on matrices$*:M(r,s;X)arrow M(s,r;X)$

by $applying*to$ the coefficients of the matrix and then taking its transpose.

The standard model form and the projective model For what follows, we set$I_{n,1}=(\begin{array}{llll}1 0 000 1 00| | \ddots ||0 0 010 0 0-1\end{array})$,

and call this the standard $fom$ . More to the point, associated tohermitian, or quaternionic hermitian) form
$B_{n,1}(x,y)=y^{*}I_{n,1}x$$X$$X^{n+1}$

B_{n,1}(x,x)<0\}$

The -projectivization of V, namely the set of
equipped with a metric$B_{n,1}$, we define the set.

-negative -lines$d([x], \beta])=\cosh^{-1}(\frac{1}{2}\frac{B_{n,1}(x,y)B_{n,1}(y,x)}{B_{n,1}(x,x)B_{n,1}(y,y)})$The metric space
space by .
$H_{X}^{n}$$(\mathcal{L}_{X}^{n},d)$is the (bilinear,,where $x,y\in X^{n+1}$ are viewed as column vectors. On$V=\{x\in X^{n+1} :$I_{n,1}$$X$$L_{X}^{n}$, can be.

is called X-hyperbolic -space and we denote this metric$n$62Arithmetic lattices and weak spectral geometry
Isometry groups Associated to$B_{n,1}$(or$I_{n,1}$) is the real Lie group$SU(B_{n,1};X)=\{A\in M(n+1;X)$:$I_{n,1}^{-1}A^{*}I_{n,1}A=I_{n+1}\}$.

The identity component of the associated projective group
acts onand leaves invariant. It is a simple matter to see that $PSU(B_{n,1};X)$ preserves the metric upon noting that for all $x,y\in X^{n+1}$ , the elements of $SU(B_{n,1};X)$
are precisely those linear transformations such that
$PSU(B_{n.1,\prime};X)$$\mathbb{P}X^{n}$$l_{X}^{n}$

$d$

$A$

$B_{n,1}(Ax,Ay)=B_{n,1}(x,y)$.

The group $PSU(B_{n,1};X)$ is, up to finite index, the full $isome\alpha y$ group of thespace . For notational simplicity, we use the $\alpha aditional$ notation:$me\alpha ic$$H_{X}^{n}$$PSU(B_{n,1};R)=PSO(n, 1)$$PSU(B_{n,1} ; C)=PSU(n, 1)$$PSU(B_{n,1};\mathbb{H})=PSp(n, 1)$.

Lattices and manifolds Given a torsion free, discrete subgroup of Isom(F),is Riemannian manifold which is locally isometric to
the quotient. We
volume,call such manifolds X-hyperbolic manifolds. Whenhas finite
we
say is a lattice and if in additionsay
weis compact,
is cocompact. According to the
Rigidity Theorem (see [37] and [45]), there is a bijection
between the isometry classes of finite volume -hyperbolic -manifolds and the-conjugacy classes of lattices in Isom(F). Consequently, to understandIsom
the former it suffices to understand the latter and we will only be concemed withup to wide commensurability. Recalllattices in Isomare commen$g\in
G$
in
wide
sense
if
for
surable thesome
and $j=1,2$ .
$\Gamma$$H_{X}^{n}/\Gamma$$H_{X}^{n}$$H_{X}^{n}/\Gamma$$\Gamma$$\Gamma$$II_{X}^{n}/\Gamma$$S\theta ong$

$X$

$n$

$(H_{X}^{n})$$\Gamma_{1},\Gamma_{2}<G$$(II_{X}^{n})$$[\Gamma_{j} :g-1\Gamma_{1}g\cap\Gamma_{2}]<\infty$

2 Arithmetic constructions
In this section, we introduce a general construction for lattices in Isom
. Before commencing with this task, we provide an overview on nonarithmetic manifolds. In the case of , nonarithmetic lattices exist in every dimension (see [18]).
However, in high dimensions, these manifolds are hybrids arising from gluing pairsof carefully chosen arithmetic ones along totally geodesic hypersurfaces. In thecase of , for $n>1$ , every lattice is arithmetic by rigidity theorems of Corlette, the story is far less complete.
[10] and Gromov-Schoen [17]. In the case of
Nonarithmetic lattices are known to exist when $n=2,3$ by work of Mostow [38]and Deligne-Mostow [11]. $ln$ higher dimensions it is unknown whether or not$(H_{X}^{n})$$II_{R}^{n}$$II_{H}^{n}$$H_{C}^{n}$63
Arithmetic lattices and weak spectral geometry
nonarithmetic lattices exist. With this said, we hope that this section will provide
those interested but not familiar with arithmetic constructions some basic knowledge on constructing arithmetic lattices. A more detailed introduction can be foundin [59].

2.1 Two basic arithmetic examples. The quotient
The first example of an arithmetic lattice is the subgroupwith the geometry inducedis the standard flat -torus (upon equipping
provides us with another example,
ffom the standard inner product). The latticenamely the subgroup $SL(n;Z)<SL(n;R)$ of those elements of $SL(n;R)$ which
preserve . To be complete, we must say in which sense this is a lattice, and thisis done as follows. We can equip $SL(n;R)$ with a volume form which is invariant
inunder both left and right translation in $SL(n;R)$ . For instance, if we select$\Lambda^{dimSL(n;R)}T_{id}SL(n;R)$ , a volume form on the tangent space of $SL(n;R)$ at thein $\Lambda^{\dim(SL(n;R))}T_{g}SL(n;R)$ to be the image ofidentity element, we define
under the map induced by the isomorphism$Z^{n}\subset R^{n}$$R^{n}/Z^{n}$$R^{n}$$n$$Z^{n}$$Z^{n}$$\omega$$\mathfrak{B}d$$\Re d$$\omega_{g}$$dR_{g}-1:T_{g}SL(n;R)rightarrow T_{id}SL(n;R)$

,

where
is the diffeomorphism of $SL(n;R)$ given by right multiplication by. The volume form provides $SL(n;R)$ with a measure via integration andas it is invariant under $SL(n;Z)$ , descends to a measure on the quotient space$SL(n;R)/SL(n;Z)$ . It is with respect to this measure that the quotient space$SL(n;R)/SL(n;Z)$ has finite volume.
$R_{g}-1$$g^{-1}$$\omega$More generally, if is a locally compact topological group equipped with a rightcomes
Haar measure , for any discrete subgroup of , the quotient space
has
if
is
a lattZce
equipped with the induced quotient measure.1 We say
is compact, we say
finite volume with respect to this measure. If in addition
modulo a maximal
is the coset space of Isomis a cocompact lattice. As
compact subgroup $K$, this definition and the one given above specific to Isomcoincide. This identification on the level of sets is made by using theand the fact that point stabilizers are maximal compact
onaction of Isom.
subgroups of Isom$G$$\Gamma$

$\mu$

$G$

$G/\Gamma$$\Gamma$$G/\Gamma$$G/\Gamma$$\Gamma$$(II_{X}^{n})$$II_{X}^{n}$$(H_{X}^{n})$$\alpha ansitive$$(H_{X}^{n})$$H_{X}^{n}$$(H_{X}^{n})$1A right Haar measure is a regular Borel measure on which is invariant under the right action$G$

on itself. It is well known that every locally compact topological group admits a right Haar
measure.

of

$G$

64
Arithmetic lattices and weak spectral geometry

2.2 Lattices arising from forms
We start the generalization of the above pair of examples to Isomwith perhaps
the most elementary construction based on bilinear, hermitian, and quaternionichermitian forms over $R,C$ , and , respectively. We call this construction the $fom$
construction).
construction (for $X=C$ , we sometimes refer to this as the first$(H_{X}^{n})$$\mathbb{H}$$\eta pe$

Model forms and the basic examples We say $B\in GL(n+1;X)is*$ -symmetric
if $B=B^{*}$ and say that $a*$ -symmetric matrix $B\in GL(n+1;X)$ is a model $fom$ if
has signature pair $(n, 1)$ . That is, upon diagonalizing , all the eigenvalues are real
, we say thatand precisely of the eigenvalues are positive. For a subringis R-defined if can be conjugated into $GL(n+1;R)$ .
which is -defined for any subring
The simplest example of a model form isof containing Z. Setting
$B$

$B$

$R\subset X$

$n$

$B$

$B$

$R$

$I_{n,1}$

$R$

$X$

$0_{X}=\{\begin{array}{ll}Z, X=R,Z[i], X=C,Z[i,j,k], X=\mathbb{H},\end{array}$. The
by work of $Borel-Harish$-Chandra [4], $PSU(n, 1;t9_{X})$ is a lattice in Isom
proof of this takes the only possible route, consffucting a finite volume fundamental. Actually, one constructs a fundamental
set for the action of $PSU(n, 1;t9_{X})$ on
using reduction theory (see [44]).
set for the action of $PSU(n, 1;t9_{X})$ on Isom$(H_{X}^{n})$$II_{X}^{n}$$(H_{X}^{n})$More generally, for any
Lie group$0_{X}$-defined model form

$B$

in $GL(n+1;X)$ , we have a real

$PSU(B;X)=\{A\in SL(n+1;X) : B^{-1}A^{*}BA=I_{n+1}\}$

with subgroup $PSU(B;O_{X})$ . Selecting a real analytic isomorphism between $PSU(B;X)$
and $PSU(n, 1;X)$ (one can take this to be conjugation in $GL(n+1;X)$ ), the image$oPSU(B;0_{X})$ is a lattice in Isom.
$(H_{X}^{n})$One interesting side note is that for $X=R$ , ranging over all the possible forms ,the above $cons\alpha uction$ produces infinitely many distinct wide commensurability
classes of lattices. While for $X=C$ or , this produces one wide commensurability
class. To produce additional wide commensurability classes over and , one.
must change the ring
Using work of Kneser (see [41]), $Borel-Harish$-Chandra [4], and Mostow-Tamagawa, for all[36], the lattices $PSU(B, t9_{X})$ are noncocompact for all
when
when $X=C$ and $n>1$ , and for all when $X=R$ and $n>3_{:}$ In particular, we have.
not yet found a construction for producing cocompact lattices in Isom$B$$\mathbb{H}$$C$$\mathbb{H}$$O_{X}$$B$$X=\mathbb{H}$$B$$(H_{X}^{n})$

$B$

65

Arithmetic lattices and weak spectral geometry
Cocompact examples in $PSO(n, 1)$ For a finite field extension
up to the field isomorphisms of and , finitely many embeddings$k/\prime Q$$R$$\sigma_{1},$$\ldots,$$\sigma_{r_{1}}$, there are,$C$

: $karrow R$ ,$\tau_{1},$$\ldots,\tau_{r_{2}}$: $karrow C$

where for the latter we insist thatnot be contained in R. For instance, when$k=Q(\sqrt{2})$ , we have a pair of embeddings which we identify with theelements
of $Ga1(Q(\sqrt{2})/Q)$ . We say is totally real if $r_{2}=0$ and totally imaginary if$r_{1}=0$ . Moreover, given a totally real extension $F$ of $Q$ , by adjoining
to$\tau_{j}(k)$$k$$\sqrt{-d}$$F$

where $d\in N$ is square-Ree, we (generically) obtain a totally imaginary quadratic
extension $E/F$ of . We call the pair $E/F$ a CM field (CM stands for complex
multiplication). Finally,shall denote the ring of algebraic -integers.
For a totally real field , fix an embedding : $karrow$ R. For a k-&fined model, we obtain a new form
form $B\in GL(n+1;k)$ , for eachwith signaturepair
by applying to the matrix . We say that is admissible if
$F$

$k$

$0_{k}$$k$$\sigma_{1}$$\sigma_{JB}$$\sigma_{j}\neq\sigma_{1}$$(p_{j},q_{j})$

$B$

$B$

$\sigma_{j}$$(p_{j},q_{j})=(n+1,0)$

for all $j\neq 1$ . Again, work of $Borel-Harish$ -Chandra implies that $PSO(B;O_{k})$ is alattice in Isom.
$(H_{R}^{n})$Example. For $k=Q(\sqrt{2})$ , we can take

$B$

to be

$B=(\begin{array}{lllll}1 0 0 00 1 0 0| | \ddots \vdots \vdots 0 0 1 \vdots 0 0 0 -\sqrt{2}0\end{array})$

For the nontrivial Galois involution$\sigma$.

, the resulting form$\sigma_{B=}(\begin{array}{lllll}1 0 0 00 l 0 0| | \ddots | |0 0 l 00 0 0 \sqrt{2}\end{array})$

is positive definite as required.
Indeed, for any totally real field $F$ , the Weak Approximation Theorem allows forthe selection of$\alpha_{1,\ldots,\%+1}\in t9_{F}$such that$B=diag(\alpha_{1},\ldots, \%+1)$

is admissible. It follows Rom $Borel-Harish$ -Chtdra and Mostow-Tamagawa thatthese lattices $PSO(B;0_{F})$ are cocompact for any $F\neq Q$ .

66Arithmetic lattices and weak spectral geometry

Cocompact examples in $PSU(n, 1)$ For $X=C$ , we can take a CM field $E/F$ andselect a model form defined over which is admissible. Viewing instead as ahermitian matrix and taking instead the associated group $PSU(B;C)$ , the subgroup$PSU(B;0_{E})$ is a lattice in $PSU(n, 1)$ and is cocompact so long as $F\neq Q$ .
$F$

$B$

$B$

Cocompact examples in $PSp(n, 1)$ To produce lattices in $PSp(n, 1)$ , we need
, we definesome new algebraic objects. For a totally real field and$F$$\alpha,\beta\in F$$A_{\alpha,\beta}=( \frac{\alpha,\beta}{F})$to be the -dimensional -algebra spanned by 1, $x,y,xy$ (as a -vector space) withmultiplication given by$F$$x^{2}=\alpha$

for allbedding$\lambda\in F$$\sigma_{j}$,$F$$y^{2}=\beta$. The algebra

of

$F$

,

$\eta=-yx$ ,$\lambda x=x\lambda$,$\lambda y=y\lambda$

is called a F-quatemion algebra. For each eminto $R$ , we obtain a new algebra$A_{\alpha,\beta}$$\sigma_{jA\otimes_{F}R=}(\frac{\sigma_{j}(\alpha),\sigma_{j}(\beta)}{R})$,and according to a theorem of Wedderbum (see [43]),$\sigma_{j}A\otimes_{F}R\cong \mathbb{H}$or $M(2;R)$ .

for all . For a model for
have the property that,say
$B\in GL(n+1;A)$ we
is admissible as before if the signature pair for all$j\neq 1$ is $(n+1,0)$ .
Taking $a,\beta\in O_{F}$ , we have the subring $0=0_{F}[1,x,y,\eta]$ ,$SU(B;0)$ is a lattice in $PSp(n, 1)$ by work of Borel-Harish-Chtdra.

We require$A$$\sigma_{jA}\otimes_{F}R\cong \mathbb{H}$

$j$

$B$

Remark. Up to wide commensurability, one can take to reside in $GL(n+1;F)$).
(indeed, can be assumed to be diagonal with coefficients in
$B$

$B$

$O_{F}$

2.3 Arithmetic constructions in general.
In this short subsection, we give a quick overview on arithmetic lattices in IsomIn particular, we mention how typical the above examples are and wben there exist
additional consffuctions of arithmetic lattices.
$(H_{X}^{n})$$InPSp(n, 1)$$Thelatticescons\alpha uctedaboveyieldalla\dot{n}thmeticlatticesinPSp(n, 1)$

up to wide commensurability so long as
isometry between$H_{R}^{4}$

and$H_{Eh}^{1}$.

$n\neq 1$. In this exceptional case, there is an67Arithmetic lattices and weak spectral geometry
In $PSO(n, 1)$ For $n+1$ odd, this produces all thelattices in $PSO(n, 1)$ .
For odd and not equal to 3 or 7, there is but one other construction in $PSO(n, 1)$
which utilizes quatemion algebras. The case of $n=3$ is exceptional due to a local
isomorphism between $SO(3,1)$ and $SL(2;C)$ . In the case $n=7$ , there is anotheranithmetic construction coming Rom
algebras. This construction is possible
due to an unusually large symmetry group for the associated Dynkin diagram forthe associated complex simple Lie group.
$a\dot{n}th\iota netic$$n$$\alpha iality$

In $PSU(n, 1)$ For $X=C$ , each pair $r,d\in N$ such that $rd=n+1$ has an associatedarithmetic consoeuction. The pair $r=n+1$ and $d=1$ is the one given above andproduces the arithmetic lattices of first type. For the pair $r=1$ and $d=n+1$ , theconstruction utilizes cyclic division algebras over CM fields $E/F$ equipped withan involution of second kind. Essentially nothing is known about the associatedcomplex hyperbolic manifolds produced by these lattices (see [29] and [55] forsome recent work on these lattices); perhaps the deepest result is the vanishingof first cohomology for congruence covers of the associated arithmetic manifolds(see [51]). One such example is Mumford’s fake $CP^{2}[39]$ (see also [47]), which. For brevity, we have chosen to omit a
has the same rational homology as
detailed description of these constructions and refer the reader to our prelminary
manuscript [31] on this topic.
$A$$CP^{2}$

2.4 Why care about arithmetic and nonarithmetic construcbons?
It is natural to ask why one should care about arithmetic and nonarithmetic constructions. Or more to the point, why arithmetic constructions produce amenable
examples for geometers to work with. Here is a loose summary of”properties”
typical arithmetic and nonarithmetic lattices and manifolds possess:
Arithmetic$\bullet$

Predictable nature of group elements; see for instance $Cooper-Long$-Reid[9].

$\bullet$$\bullet$

Predictable nature of totally geodesic submanifolds and geodesics; see for
instance Maclachlan-Reid [26].
$Symme\alpha y$; see for instance Farb-Weinberger [12] and $Cooper-Long$-Reid[9].
$\bullet$Downside; it is difficult to find an explicit description like a group presenta-tion (see [25]).

68

Arithmetic lattices and weak spectral geometry

Nonarithmetic:$\bullet$$\bullet$$\bullet$Margulis dichotomy (see below or [28], [59]); see for instance Step 3 below.

UsuaUy ”explicitly” constructed.
Downside; Substructures (like totally geodesic submanifolds) are more mys-

terious.
$\bullet$Downside; For most $symme\alpha ic$ spaces, only arithmetic consmctions are

possible (see [28] or [59]).

3 Spectral geometry
Associated to any Riemannian -manifold $M$ are several sets which encode some
portion of the geometry and topology of $M$ . Perhaps the most natural (from ageometric viewpoint) of these sets is the geodes length spectrum consisting of
the lengths of the closed geodesics 7 on $M$ , where each length is counted with. We could instead insist that the geodesics
multiplicity. We denote this set by
be primitive or simple and this produces the primitive geodesic length spectrum and, respectively.
simple geodesic length spectrum which we denote bysets,
we call the resulting set the (primitive
If we forget the multiplicities of theseor simple) geodesic length set and denote it by $L(M)$ (resp., $L_{p}(M),L_{s}(M)$ ).
$n$

$ic$

$\mathcal{L}(M)$$l_{p}(M),\mathcal{L}_{s}(M)$Another natural set to associate to $M$ is the spectrum of the Laplace-Beltrami operator acting on the Hilbert space $L^{2}(M)$ of square integrable functions of $M$ . Moregenerally, this operator acts on the Hilbert space of square integrable -forms andand. Given
we denote the spectra for this operator on these spaces by
$M,N$
,
among
we have equality
thea pair of isometric Riemannian -manifolds
iftheholds.
converse
problemso-calledinverse
asks
The
the
pair.
spectra for$\epsilon(M)$$\mathcal{E}_{p}(M)$$n$Question (Inverse Problem). If $\epsilon(M)=\epsilon(N)$ , are $M$ and $N$ isometric?

In 1964, Milnor [33] answered this question in the negative by producing a pairof nonisometric flat 16-tori with equal eigenvalue spectra. Since Milnor’s article,
many additional examples have been given. Most notably for us is a construction
due to Sunada [56] which is purely algebraic. For brevity alone, we shall speak
in detail only on this consffuction and refer the reader to the survey [15] for an
detailed overview on the subject of isospectral $cons\alpha uctions$ .

69Arithmetic lattices and weak spectral geometry

3.1 Sunada’s method
Sunada’s construction (which itself was inspired by number theory; see [42]) utilizes the following group theoretic concept.

Definition 1. For afinite ハク uP , we call apair of subgroup $H,K$ ahmmost conjugate zffor each G-conjugacy class $[g]$ , we have the equality
$G$

$g$

$|H\cap[g]|=|K\cap[g]|$.

For a Riemannian -manifold $M$ whose fundamental group $\pi_{1}(M)su\dot{\eta}ectsG$, itis an easy exercise to verify $L(M_{H})=\mathcal{L}(M_{K})$ for the menic covers corresponding
to the pullbacks of $H$ and . That these covers also have equal eigenvalue spectra
follows Rom the equivalence of almost conjugacy with the following condition.
$n$

$K$

$(\phi)$For every finite dimensional complex representation$p:Garrow GL(n;C)$

we have the equality$dimFix(p(H))=dimFix(p(K))$ .

Given the equivalence of Definition 1 and (J), it is not difficult to prove that$\epsilon(M_{H})=\epsilon(M_{K})$.

3.2 Using unada’s method$S$

Using known examples of almost conjugate pairs $H,K$, Sunada [56] produced
many new examples of isospectral, nonisometric hyperbolic 2-manifolds. Since
then, examples of isospectral hyperbolic -manifolds for every were found (see[3], [7], [49], [27], [57]). For complex and quatemionic hyperbolic manifolds,
Spatzier [53] (see also [54]) found examples so long as the dimension is sufficiently high. Recently, we completed his work [30], finding examples in every
dimension. Both of these $cons\alpha uctions$ utilize Sunada’s method. Indeed, the work
involved in applying Sunada’s method is showing the manifolds are nonisometric.
The main tool we use for this is recent work of Belolipetsky-Lubotzky [2]. Briefly,
the main points of our construction are:
$n$

$n$

with pairwise almost conjugate, noncon. Important here is that tends to infinity as a

(Step 1) Find families of finite groups

jugatefunction of .

$sub_{\Psi}oups\{H_{j,k}\}_{k=1}^{r_{j}}$$j$$N_{j}$$r_{j}$$r_{j}$

70
Arithmetic lattices and weak spectral geometry(Step 2) For a manifold $M$, find surjective homomorphisms $\pi_{1}(M)arrow N_{j}$.

(Step 3) Find bounds on the number of ways a given cover can be isomekic to anothercover ofto$N_{j}$.

$M$

associated to the pullbacks $ofH_{j,k}$ under the sur ections of$\pi_{1}(M)$It is worth noting that our approach was inspired by the approach taken by Belolipetsky and Lubotzky [1] in the resolution of the inverse Galois problem for$isome\alpha y$ groups of closed hyperbolic-manifolds. As a somewhat lengthy side
note, we describe this philosophy employed in [1].
$n$One approach to the inverse Galois problem for Riemann surfaces is as follows$[22]$ for the inverse Galois problem for hyperbolic(see [16] for rigorous
3-manifolds, and [24] for the general case of the trivial group). Using the largenessof surface groups, given a finite group , one can find a surjective homomorphism, one obtains a hyperbolic. For each hyperbolic structure on
structure on the cover corresponding to the pullback of the trivial group under
the surjection of
onto . In particular, these hyperbolic $s\alpha uctures$ always
have as a subgroup of their isometry groups; this provides an embedding of theTeichm\"uller space of
into the Teichm\"uller space of the cover. Loosely, when
the hyperbolic structure possesses more symmetry than , these $s\alpha uctures$ sit on
an embedded copy of the Teichmuller space of a smaller surface. In particular,
have precisely for their isometry
generic $s\alpha uctures$ on the image of Teich$\alpha ea\mathfrak{a}nent,$$G$$\pi_{1}(\Sigma_{9})arrow G$$\pi_{1}(\Sigma_{9})$$\pi_{1}(\Sigma_{g})$

$G$

$G$

$\Sigma_{g}$

$G$

$G$

$(\Sigma_{g})$group.

Belolipetsky-Lubotzky [1] proceed in a similar manner to produce hyperbolic $n-$
manifolds with isometry group . The real and obvious sticking point is the lackof a Teichm\"uller space due to Strong Rigidity. The variational method in their approach $kcomes\cdot discrete$ ; they produce covers of a large, nonarithmetic manifold
and by a counting argument show that some (generically) of these covers must haveprecisely for their $isome\alpha y$ group.
$G$

$t$

$G$

Sunada [56] (see also [5]) takes a similar approach for symmetry groups but insteadto produce isospectral, nonisometric Riemann surfaces. Va largeness of, wherepossesses an
one is afforded surjective homomorphisms$H,K$
space forcopiestheTeichmiiller. This produces twoalmost conjugate pairin the TeichmUller space of the surface corresponding to $H$ (or equivalent ). By
selecting a hyperbolic metric on with trivial isometry group, which by the same
reasoning above, occurs generically, the lifted metrics on the covers corresponding
to the pullbacks of $H,K$ are nonisometric (and by Sunada’s theorem, isospectral).
$\pi_{1}(\Sigma_{8})$$\pi_{1}(\Sigma_{g})arrow G$.

$G$$\Sigma_{g}$$K$$\Sigma_{9}$apWith this view, our approach in [30] was to replace the continuous
proach of Sunada with a discrete variational approach as done by Belolipetsky$V\dot{\bm{t}}ational$

71

Arithmetic lattices and weak spectral geometry
Lubotzky. The first two steps aim to produce large families of isospectral covers
while the third step replaces the Baire category argument used in a continuous
variational approach.

3.3 A sketch of how to achieve the basic steps
For completeness, we provide a sketch of how the ffiee steps to our approach areachieved.
Step One The starting point for our approach (aside from Sunada’s paper [56])is a paper of Brooks, Gornet, and Gustafson [5].
For any field , we define the 3-dimensional Heisenberg group over to be
$k$

$k$

$\mathfrak{N}_{3}(k)=\{(\begin{array}{lll}1 x t0 1 y0 0 1\end{array})$: $x,y,t\in k\}$ .

Via the inclusion of $GL(3;k)$ into $GL(n+3;k)$ into the upper three by three block,we may viewas a subgroup of $GL(n+3;k)$ for all $n\geq 0$ . The horizontal
subgroup$\mathfrak{N}_{3}(k)$$H(k)=\{(\begin{array}{lll}1 x 00 l 00 0 1\end{array})$: $x\in k\}$. Specifically, for a finite field
and twists of it will produce the sought afterwith $q=p^{n},$ $Brooks-Gomet-Gustafson[5]$ found large (depending on andn) collections of pairwise almost conjugate, nonconjugate subgroups of the fiby ”twisting” the horizontal subgroup $H(F_{q})$ by certain maps
nite groups$f:F_{q}arrow F_{q}$ . Recall thatis simultaneously an -dimensional -vector space
and a l-dimensional -vector space. The set of -linear endomorphisms is a-vector space with the -linear endomorphisms sitting as an -linear subspace. Upon selecting an -basis, the former may be identified with $M(n;F_{p})$and the latter with . The quotient -vector space $AL(F_{q})$ of $M(n;F_{p})$ by
will be called the space of twist maps. For simplicity in what follows, we flx asplitting$H_{j,k}$$p$$F_{q}$$\mathfrak{N}_{3}(F_{q})$$n$$F_{q}$$F_{p}$$F_{q}$$F_{p}$$F_{q}$$F_{p}$$F_{p}$$F_{p}$$F_{q}$$F_{p}$$M(n;F_{q})=F_{q}\oplus AL(F_{q})$which exists by elementary linear algebra.

$F_{q}$

72
Arithmetic lattices and weak spectral geometry

Given agroup

$F_{p}$-linear endomorphism $f$ of$f_{H(F_{q})}$

to be$F_{q}$, we define the f-twisted horizontal sub-$f_{H(F_{q})=}\{(\begin{array}{lll}1 x f(x)0 1 00 0 1\end{array})$:$x\in \mathbb{F}_{q}\}$.

The following lemma is due to $Brooks-Gomet-Gustafson[5]$ .

Lemma 3.1. For any pair of -linear endomorphism $f,g$, the subgroups
and $gH(F_{q})$ are almost conjugate inand conjugate in
if and only$f_{H(F_{q})}$$F_{p}$$\mathfrak{N}_{3}(F_{q})$$lff-g\in \mathbb{F}_{q}$$\mathfrak{N}_{3}(F_{q})$.

An immediate consequence of Lemma 3.1 is the existence of $p^{n(n-1)}$ pairwise almost conjugate, nonconjugate subgroups {of.
$f_{H(F_{q})\}_{f\in AL(F_{q})}}$$\mathfrak{N}_{3}(F_{q})$Step Two The resolution of Step 2 is on the one hand a formal matter, appealing
to well known results Rom number theory and the sffucmre theory of algebraic
groups. On the other hand, it is the most technical step in our approach. For thisreason, we have opted to omit a lengthy discussion of how this is achieved. The
main points are:$\bullet$$\bullet$

The Strong Approximation Theorem (see [40] and [58]).
Existence of algebraic -forms $G$ of the complexification ofmodel semisimple group with certain properties; for instance $G$ is an inner form andhas certain desired properties.
$F$

$G$

$\bullet$Ensuring that the groups

$F$

$G$

contain Heisenberg groups.

Step Three The resolution of Step 3 splits naturally into two cases. Having
achieved Steps 1 and 2 for a manifold $M$ , we split our considerations into twocases depending on whether or not $M$ is arithmetic. In the case $M$ is nonanithmetic,extremely good bounds on the number of ways finite covers of $M$ can be isometric
are obtained from deep work of Margulis [28]. In the case $M$ is arithmetic, we
appeal to work of Belolipetsky-Lubotzky [2].

4 What do the multiplicities see?
Though the $isome\alpha y$ type of a manifold is not preserved under isospectrality, certain quantities like volume and dimension are when passing between isospectral
manifolds. One basic question that can be asked is:73Arithmetic lattices and weak spectral geometry
Question. How much geometric information is encoded in the mulriplicities? For
example, is volume an invanant of the spectral set without multiplicities?

In [52], Schmutz produced infinitely many pairs of finite covers of the modular
quotient $II_{R}^{2}/PSL(2;Z)$ with identical geodesic length sets but with different volume (thus producing a negative answer to the second part of the above question).
The proof utilized the structure of $PSL(2;Z)$ , using some elementary matrix calculations; in particular, it is not a method which appears to be easy to generalize toother settings or even other lattices in $PSL(2;R)$ . Recently, with Leininger, Neumann, and Reid [23], we investigated this question and specifically the question of
how abundant such examples are. Here are some of our results:

Theorem 4.1 ([23]). Let $M$ be a closed X-hyperbolic n-manifold. Then there
exists an infinite family offinite covers
of $M$ such that$(M_{j},N_{j})$(1) $L_{p}(M_{j})=L_{p}(N_{j})$ ,(2) $vol(M_{j})/vol(N_{j})$ is unbounded as afiznction of .
$j$

Theorem 4.2 ([23]). Let $M$ be a closed X-hyperbolic n-manifold. Then there
exists an infinite family offinite covers
of $M$ such that$(M_{j},N_{j})$(1) $E(M_{j})=E(N_{j})$ ,(2) $vol(M_{j})/vol(N_{j})$ is unbounded as afimction of $j$ .

These results are achieved with variations of Sunada’s method. Below, we briefly
describe the group theoretic conditions.
It is not too difficult to show that two Riemannian manifolds with identical geodesic
length spectra do indeed have identical primitive geodesic length spectra. Moreover, for compact locally symmetric manifolds, the eigenvalue spectrum is known
to determine the primitive geodesic length spectrum, at least up to multiplication
by rational numbers (see [46]). For negatively curved manifolds, there is a evenstronger relations between the eigenvalue and primitive geodesic length spectra(see [13]), and for Riemannian surfaces, one can recover each from the other (see[19], [20], [6]). It might then come as a surprise that these implications typically
fail upon forgetting the multiplicities. Specifically, in [23], we construct examples(typically Riemann surfaces) with the following properties:$\bullet$$\bullet$$L(M_{1})=L(M_{2})$but $L_{p}(M_{1})\neq L_{p}(M_{2})$ .

$L(M_{1})=L(M_{2})$but $E(M_{1})\neq E(M_{2})$ .

74Arithmetic lattices and weak pectral geometry$s$

but $L_{p}(M_{1})\neq L_{p}(M_{2})$ .

$\bullet$$E(M_{1})=E(M_{2})$$\bullet$$E(M_{1})=E(M_{2})$ but $L(M_{1})\neq L(M_{2})$.

Of course, one always has the implication that when $L_{p}(M_{1})=L_{p}(M_{2})$ , then. Thus the only remaining relation is whether or not the equality$L_{p}(M_{1})=L_{p}(M_{2})$ implies the equality $E(M_{1})=E(M_{2})$ . There seems to be noreason to expect this to either hold or fail.

$L(M_{1})=L(M_{2})$Using examples constructed in [8], one can produce examples of closed hyperbolic 3-manifolds
with $L_{s}(M_{1})=L_{s}(M_{2})$ with arbitrarily large volume gap.
However, this does not address how much geometric content is encoded in the simple length set as the manifolds$j=1,2$ , have the remarkable property that any
possesses only simple closed geodesics. Heuristimanifold commensurable tocally, one expects closed geodesics on an -hyperbolic -manifolds to be simple
generically, so long as the manifold is not a Riemannian surface.2 This leads us toa pair of questions which we view as fundamental:$M_{1},M_{2}$$M_{j},$$M_{j}$$X$

Question. Do there exist distinct Riemann suffaces$L_{s}(X_{1})=L_{s}(X_{2})$ ?
Question. Do there exist distinct Riemann surfaces?

$n$$X_{1},X_{2}$such that$X_{1},X_{2}$such that$L_{s}(X_{1})=\mathcal{L}_{s}(X_{2})$In the latter case, it is not immediately obvious thatare homeomorphic.
However, using known asymptotic upper and lower bounds on the number of sim,ple closed geodesics on a Riemann surface ([34], [35]), it follows that
topologically. One reason to perhaps expect more geometric content in the simple
length spectrum is the fact that one can determine the Riemann surface knowing
only the length of a special finite collection of closed curves on the surface. Nevertheless, it seems too early to conjecture simple length spectral rigidity for Riemannsurfaces.
$X_{1},X_{2}$$X_{1}\cong X_{2}$To the author’s knowledge, equality of simple geodesic length sets is not known to
imply that the surfaces are topologically equivalent. Rivin [50] has conjectured thatthe multiplicities in the simple geodesic length spectrum are bounded (independent of the hyperbolic structure); the multiplicity is known to be one for a genericsurface by a straightforward Baire category argument (see for instance [32]). If
Rivin’s conjecture holds, then the simple geodesic length set would detemine thetopological type by again appealing to the asymptotic growth rate of simple closed

2At present it is unknown whether or not every finite volume hyperbolic -manifold possesses
infinitely many simple closed geodesics up to free homotopy.
$n$

75

Arithmetic lattices and weak spectral geometry

geodesics. Indeed, we only require that the multiplicities in the spectrum be relatively small in comparison to the number of simple closed curves.
Remark. For flat tori, despite the fact that simple multiplicity need not be bounded(see [32]), one can find linear bounds on the simple multiplicities as a function oflength. Indeed, one can make a coarse geometric argument using the isoparametricinequality forto see this. It seems plausible that even if Rivin’s conjecture isfalse that one might be able to produce polynomial bounds on the simple multiplicity as a function of length.
$R^{2}$

Finally, for length, primitive, and simple geodesic length sets, the number of pairwise distinct surfaces of genus which can be pairwise length, primitive, or simple
geodesic length equivalent is finite. Indeed, by continuity of length such a set isdiscrete in the moduli space of genus curves and contained in a compact set ofby Mumford’s compactness criterion.
$g$

$g$

$M_{g}$

5 Using symmetry in spectral constructions: Sunada’s method
and some variants
In the next three subsections, the associated Sunada-type group theoretic condition
will be given for length, eigenvalue, and primitive length set equivalence.

5.1 Elementwise$co\iota\dot{u}ugate$

Our first definition is motivated from Definition 1.

Definition 2. Given a group (not necessarily finite) and a pair of subgroups$H,K<G$, we say $H,K$ are elementwise conjugate if$G$$\bigcup_{\epsilon\in G}g^{-1}Hg=\bigcup_{g\in G}g^{-1}Kg$

If

$G$

.

is finite, this is equivalent to:$(*)$

for all-conjugacy classes

$[g]$

,

$H\cap[g]\neq\emptyset\iota f$

and only$\iota fK\cap[g]\neq\emptyset$.

The following is one of the main examples used in [23] to produce manifolds withequal geodesic length sets (for instance examples of closed hyperbolic -manifolds
in every dimension).
$n$

76
Arithmetic lattices and weak spectral geometry

Example. Let

be an odd prime,the (unique) finite field with elements,$G=F_{p}^{n}\rtimes SL(n;F_{p}),$ $H=W$ , and $K=V$ , where
are non-trivial
subspaces. The inclusion of into provides us with a pair of subgroups $H,K$ in. The transitivity of the action of $SL(n;F_{p})$ on the set of -lines in is enoughto imply that $H,K$ are elementwise conjugate in .
$p$

$F_{p}$$p$$F_{p}-$$W,V\subset F_{p}^{n}$$G$$F_{p}^{n}$$G$$F_{p}$$F_{p}^{n}$$G$

5.2 Fixed point equivalent
Our next definition is motivated by $(l)$ .
Definition 3. We say subgroups $H$ and of a finite groupalent iffor any finite dimensional complex representation
has a nontrivial fixed vector and only
does.
$K$

$\rho|_{H}$$\iota f$$G$

arefixed point equivof , the restnction$\rho$$G$$\iota f\rho|\kappa$

It is not true that Definitions 2 and 3 are equivalent unlike the equivalence of Definition 1 and $(l)$ . This is the first indication that relationships upon forgetting
multiplicities could be more subtle.
The elementwise conjugate examples above also produce fixed point equivalentpairs with a slightly different condition on the subspaces $V,W$ .
Example. With $G=F_{p}^{n}\rtimes SL(n;F_{p})$ , if $H=W$ , and $K=V$ , whereare proper -subspaces, then $H,K$ are fixed point equivalent subspaces of .
The proof of this uses standard results Rom character theory in tandem with anelementary counting argument.
$W,V\subset F_{p}^{n}$$G$$F_{p}$

5.3 Primmitive pairs
Our final group theoretic concept does not fit into the general pattem taken withthe previous two. Nevertheless, this condition does produce manifolds with equalprimitive geodesic length sets.
theDefinition 4 (Primitive). We shall call a subgroup $H$ of primuive infollowing holds:
$G$

$(a)$

All non-tnvial cyclic subgroups of $H$ have the same order$G\iota f$$p$(necessarily

prime).
$(b) \bigcap_{g\in G}g^{-1}Hg=\{1\}$.

As before, primitive pairs can be found in$F_{p}^{n}\rtimes SL(n;F_{p})$.

, if $H=W$ ,Example. Setting as before to be the affine groupare proper, nontrivial -subspaces, then $H,K$ are
and $K=V$ , where
primitive and elementwise conjugate; (a) is trivial to verify while (b) again follows
Rom the transitivity of the action of $SL(n;F_{p})$ on the set of -lines.
$G$$W,V\subset F_{p}^{n}$$F_{p}^{n}\rtimes SL(n;F_{p})$$F_{p}$$F_{p}$

77

Arithmetic lattices and weak spectral geometry

5.4 A variant of Sunada’s theorem
One of the main results of [23] is the following variation on Sunada’s theorem.
Theorem 5.1. Let $M$ be a Riemannian manzfold,ハ\mbox{\boldmath $\nu$}iSe conjugate subgmups of .

$G$a group, and $H$ and$K$elemen-$G$(1) If $\pi_{1}(M)$ admits a homomorphism onto $G$, then $L(M_{H})=L(M_{K})$ for thecovers$M_{H}$

andassociated to the pullback subgroups $ofH$ and$M_{K}$.

$K$(2) If, in addition, $H$ and $K$ are primitive in $G$ and $\pi_{1}(M)$ has the property thatany pair of distinct maximal cyclic subgroups of$L_{p}(M_{H})=L_{p}(M_{K})$.

$\Gamma$in tersect tnvially, then(3) If instead $H$ and $K$ are fixed point equivalent, then $E(M_{H})=E(M_{K})$ .

The reader will note that on top of being less natural in regard to the associatedgroup theoretic condition, the production of primitive geodesic length equivalentmanifolds also requires conditions on the fundamental group $\pi_{1}(M)$ of the Riemannian manifold. The condition on maximal cyclic subgroups required in our
proof is likely not needed (that some condition is required is seen from examples

in [23]).

5.5 The existence of weak spectrally equivalent covers
To prove our results in the generality stated above (i.e., for any closed hyperbolic-manifold), one can typically work with the examples of pairs $H,K$ given above.
In dimensions 3,4 however, other examples are required. These pairs are similarof a fixed abelian -group which into those given above being subgroups
for some $\theta<Aut(A)$ . The virtualtum is embedded in a semidirect productonto groups of this form follows Rom the Strong Approximasurjection oftion and Cebotarev Density Theorems. The lion’s share of the work is in showing
are primitive, elementwise conjugate, and eigenvalue equivthat these pairs
alent.
$n$$A$$A_{1},A_{2}$$A\rtimes\theta$$\pi_{1}(M)$$A_{1},A_{2}$These methods also work to produce covers over any closed -hyperbolic $n-$
manifold. In addition, one can also produce arbitrarily long towers of covers$X$$M_{r}arrow M_{r-1}arrow\ldots-M_{2}arrow M_{1}arrow M$

is length, pnimitive length, or eigenvalue equivalent.
such that each pair
These methods also work more generally for locally symmetric manifolds of noncompact type.
$M_{j},M_{k}$78Arithmetic lattices and weak spectral geometry

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Califomia Institute of Technology
Department of Mathematics
Pasadena, CA 91125
email: dmcreynQcaltech. edu