# Spectral gap and character limits in
arithmetic groups

Arie Levit, Raz Slutsky and Itamar Vigdorovich

###### Abstract.

We establish vanishing results for limits of characters in various discrete
groups, most notably irreducible lattices in higher rank semisimple Lie
groups. As an application, we show that any sequence of finite-dimensional
representations converges to the regular representation in the Fell topology.
We achieve this by studying the geometry of the simplex of traces of discrete
groups having Kazhdan’s property (T) or its relative generalizations.

##  1\. Introduction

Let $\Gamma$ be a countable group. A _trace_ on $\Gamma$ is a positive
definite, conjugation-invariant function $\varphi:\Gamma\to\mathbb{C}$
normalized so that $\varphi(e_{\Gamma})=1$. The space
$\mathrm{Tr}\left(\Gamma\right)$ of all traces on the group $\Gamma$ is convex
and compact in the topology of pointwise convergence. The extreme points of
$\mathrm{Tr}\left(\Gamma\right)$ are called _characters_. The space of all
characters is denoted by $\mathrm{Ch}\left(\Gamma\right)$. In general
$\mathrm{Ch}\left(\Gamma\right)$ need not be compact.

Characters play an important role in harmonic analysis. For example, the space
of characters of the group $\Gamma$ coincides with its Pontryagin dual when
$\Gamma$ is abelian and with the set of equivalence classes of its irreducible
representations when $\Gamma$ is finite. Recently, the study of traces on
discrete groups has led to remarkable applications towards understanding the
group structure, its dynamics, stability, rigidity and more, see e.g. [Bek07,
BH21, BBHP22, HS18, LV23, LL23].

We study the character theory of higher rank lattices in semisimple groups.
Our main result concerns the asymptotic behaviour of sequences of such
characters.

######  Theorem 1.1.

Let $G$ be a semisimple Lie group with finite center. Assume that
$\mathrm{rank}_{\mathbb{R}}(G)\geq 2$ and that some non-compact simple factor
of $G$ has property (T). Let $\Gamma$ be an irreducible lattice in the Lie
group $G$. If $\varphi_{n}\in\mathrm{Ch}\left(\Gamma\right)$ is any sequence
of pairwise distinct characters then $\varphi_{n}(\gamma)\to 0$ for all
elements $\gamma\in\Gamma\setminus\mathrm{Z}(\Gamma)$.

In particular, if the lattice $\Gamma$ is center-free then any such sequence
of characters $\varphi_{n}\in\mathrm{Ch}\left(\Gamma\right)$ converges
pointwise to the Dirac character
$\delta_{e}\in\mathrm{Ch}\left(\Gamma\right)$.

Our proof of Theorem 1.1 depends on a more general principle (Theorem 6.1).
Here are some additional situations where this principle applies, and we
establish asymptotic vanishing results for characters. The full details are
provided in §6.

  1. (1)

Higher rank $S$-arithmetic groups over global fields, provided that the
enveloping semisimple group admits a non-compact simple factor with property
(T); see Theorem 6.5.

  2. (2)

The linear groups $\mathrm{SL}_{2}(S^{-1}\mathcal{O}_{k})$ where $k$ is an
algebraic number field with ring of integers $\mathcal{O}_{k}$ and
$S\subset\mathcal{O}_{k}$ is a finite (possibly empty) multiplicative subset
such that the localization $S^{-1}\mathcal{O}_{k}$ has infinitely many units;
see Theorem 6.3.

  3. (3)

In the absence of Kazhdan’s property (T) we may rely on Selberg’s property
instead. We establish convergence of congruence characters, i.e. characters
factoring through a congruence quotient; see Theorem 6.4.

  4. (4)

Lattices in certain non-semisimple Lie groups, e.g. the group
$\mathrm{GL}_{n}(\mathbb{Z})\ltimes\mathbb{Z}^{n}$ with $n\geq 3$; see Theorem
6.6.

### Relation to invariant random subgroups

An important connection between characters and dynamics is provided by the
following notion coined in [AGV14]. An _invariant random subgroup_ of the
group $\Gamma$ is a $\Gamma$-invariant Borel probability measure on the
Chabauty space of its subgroups. The key observation is that given an
invariant random subgroup $\mu$ of the group $\Gamma$ one can construct a
trace $\varphi_{\mu}\in\mathrm{Tr}\left(\Gamma\right)$ by setting
$\varphi_{\mu}(\gamma)=\mu(\\{H\leq\Gamma\;|\;\gamma\in H\\})$ for all
elements $\gamma\in\Gamma$.

A major work dealing with invariant random subgroups of lattices in higher
rank semisimple Lie groups is [ABB+17]. Many of the striking applications of
that work are obtained by showing that every sequence of pairwise distinct
irreducible invariant random subgroups of such a lattice converges to a
measure supported on subgroups of the center. Theorem 1.1 is an analogue of
this phenomenon for characters.

We mention in passing that the special case of the convergence result from
[ABB+17] for subgroups of a given lattice can be inferred from our Theorem 1.1
via a careful analysis of traces associated to finite-index subgroups.

### Character bounds of finite groups

Note that characters of quotients lift to characters of the group itself. Here
is an application. Consider any sequence of non-trivial characters
$\varphi_{i}\in\mathrm{SL}_{n}(\mathbb{F}_{p_{i}})$ where $p_{i}\in\mathbb{N}$
are distinct primes and $n\geq 3$. A consequence of Theorem 1.1 is that
$\varphi_{i}([\gamma]_{p_{i}})\to 0$ for any fixed non-central element
$\gamma\in\mathrm{SL}_{n}(\mathbb{Z})$. Here $[\gamma]_{p_{i}}$ denotes the
entry-wise reduction of the matrix $\gamma$ modulo the prime $p_{i}$. More
generally, given a matrix $\gamma$ defined over a ring $R$ and an ideal $I\lhd
R$, we let $[\gamma]_{I}$ denote the matrix obtained by reducing modulo $I$
entry-wise. We deduce the following statement on characters of simple (as well
as non-simple) finite groups of Lie type; see p. 6 for the exact definitions.

######  Corollary 1.2.

Let $G$ be a Chevalley group of rank at least two111The Chevalley group $G$
can be one of the classical Chevalley groups $A_{n},B_{n},C_{n}$ or $D_{n}$
for some $n\geq 2$ or one of the exceptional Chevalley groups
$E_{6},E_{7},E_{8},F_{4}$ or $G_{2}$.. Let $K$ be a global field and
$\mathcal{O}_{K}$ its ring of integers. Then for any non-central element
$\gamma\in G(\mathcal{O}_{K})$ and for any $\varepsilon>0$ the bound

(1.1) $|\varphi([\gamma]_{I})|<\varepsilon$

holds for all but finitely many ideals $I$ of $\mathcal{O}_{K}$ and for any
faithful222A character $\varphi$ is faithful if $\varphi(g)\neq 1$ for every
non-trivial element $g$. If the ideal $I$ is maximal then the group
$G(\mathcal{O}_{K}/I)$ is simple so that any non-trivial character is
faithful. character $\varphi$ of the finite group $G(\mathcal{O}_{K}/I)$.

Quantitative bounds on characters of finite groups are called character
bounds. These have been extensively studied, see e.g. [Glu93, Glu97, LST11,
BLST18]. Exponential character bounds have found extraordinary applications,
most notably in the proof of the Ore conjecture [LOST10]. While our statement
is not quantitative, it does provide a relatively simple framework for
understanding why such bounds can be expected. Moreover, character bounds have
been mostly considered for finite _simple_ groups. To the best of our
knowledge, Corollary 1.2 is novel for certain non-simple groups, such as
$\mathrm{SL}_{d}(\mathbb{Z}/p^{i}\mathbb{Z})$ with $p$ prime. See also Theorem
7.7 for character bounds of
$\mathrm{SL}_{d}(\mathbb{F}_{p_{i}})\ltimes\mathbb{F}_{p_{i}}^{d}$ provided
$d\geq 3$ and with $p_{i}$ prime.

### Convergence of representations

Convergence of traces implies convergence of the corresponding representations
with respect to the Fell topology; see Lemma 5.6. Therefore Theorem 1.1 on
character limits implies convergence of representations to the regular
representation.

######  Theorem 1.3.

Let $\Gamma$ be an irreducible lattice as in Theorem 1.1 and with trivial
center. Then any sequence of pairwise non quasi-equivalent333Two finite-
dimensional unitary representations $\pi_{1}$ and $\pi_{2}$ are called _quasi-
equivalent_ if they have the same set of irreducible subrepresentations.
finite-dimensional unitary representations of the group $\Gamma$ converges to
the regular representation $\lambda_{\Gamma}$ in the Fell topology.

Theorem 1.3 applies in additional and more general situations, such as the
ones mentioned following Theorem 1.1 on page 1.

Recall that the Fell topology is not Hausdorff. In particular, limits are not
unique. Relying on [BBH21] it can be shown that _any_ limit of a sequence of
pairwise disjoint finite-dimensional unitary representations of an irreducible
lattice $\Gamma$ as in Theorem 1.3 weakly contains the regular representation
$\lambda_{\Gamma}$ (see Proposition 6.7). We do not know whether all such
limits must, in fact, be weakly equivalent to the regular representation
$\lambda_{\Gamma}$.

######  Corollary 1.4.

Let $\Gamma$ be an irreducible lattice as in Theorem 1.1 and with trivial
center. Let $\Gamma_{n}\leq\Gamma$ be a strictly descending chain of finite-
index subgroups. Then the sequence of quasi-regular representations
$L^{2}(\Gamma/\Gamma_{n})$ converges to the regular representation
$\lambda_{\Gamma}$ in the Fell topology.

Corollary 1.4 can also be deduced as a consequence of the Benjamini–Schramm
convergence of the sequence of Schreier graphs of $\Gamma/\Gamma_{n}$ to the
Cayley graph of the group $\Gamma$, which in turn follows from [ABB+17]. Note
that the quasi-regular representations $L^{2}(G/\Gamma_{n})$ of the semisimple
Lie group $G$ converge to its regular representation $\lambda_{G}$, by the
continuity of induction.

### Spectral gap and the simplex of traces

Let $\Gamma$ be a countable group. The compact and convex space of traces on
the group $\Gamma$ is a metrizable _Choquet simplex_ ,444We will sometimes use
the shorter terminology _simplex_ to refer to Choquet simplices. We refer to
[Phe01] for the general theory of simplices. i.e. every trace
$\varphi\in\mathrm{Tr}\left(\Gamma\right)$ can be expressed as the barycenter
of a _unique_ Borel probability measure supported on the set of its extreme
points $\mathrm{Ch}\left(\Gamma\right)$. A Choquet simplex is called _Bauer_
if its set of extreme points is closed (and hence compact).

Our strategy to prove Theorem 1.1 is to exploit the geometric structure of the
simplex $\mathrm{Tr}\left(\Gamma\right)$. First, we show that the simplex
$\mathrm{Tr}\left(\Gamma\right)$ is Bauer. Next, we show that every non-
central character is an isolated point of the subset of extreme points of this
Bauer simplex. These two facts immediately imply that every sequence of
pairwise distinct characters of the group $\Gamma$ must converge to a
character supported on the center.

To pinpoint closed subsets of the space of characters
$\mathrm{Ch}\left(\Gamma\right)$ we make use of the notion of property
$\mathrm{T}(\mathcal{R})$. Let $\mathcal{R}$ be a set of equivalence classes
of unitary representations of the group $\Gamma$. For each unitary
representation $\pi\in\mathcal{R}$ let $\pi_{0}$ denote its subrepresentation
obtained by taking the orthogonal complement to the subspace of invariant
vectors. The group $\Gamma$ has _property $\mathrm{T}(\mathcal{R})$_ if the
trivial representation $1$ is an isolated point of the set
$\\{\pi_{0}:\pi\in\mathcal{R}\\}\cup\\{1\\}$ with respect to the Fell
topology. In particular, if the family $\mathcal{R}$ consists of all
equivalence classes of irreducible unitary representations then property
$\mathrm{T}(\mathcal{R})$ is the same as Kazdhan’s property (T). This
generalized property was first introduced in [LZ89]. See also §5 below for
further discussion. Denote

(1.2)
$\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)=\\{\varphi\in\mathrm{Ch}\left(\Gamma\right)\>:\>\pi_{\varphi}\in\mathcal{R}\\}$

where $\pi_{\varphi}$ is the unitary representation associated to the
character $\varphi$ via the GNS construction, see §2. When discussing the
subset $\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)$ we will be making a very
mild assumption, namely that the family $\mathcal{R}$ is so called
_hereditary_ , a technical notion defined in §5. The first ingredient towards
Theorem 1.1 and its generalizations is the following.

######  Theorem 1.5.

Assume that the countable group $\Gamma$ has property
$\mathrm{T}(\mathcal{R})$. Then
$\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}\subset\mathrm{Ch}\left(\Gamma\right)$.
Moreover, the closed convex hull
$\overline{\mathrm{conv}\,\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$ is a
closed face of the simplex $\mathrm{Tr}\left(\Gamma\right)$ and is a Bauer
simplex.

The second part of Theorem 1.5 follows from its first part purely abstractly.
Specializing to the case where the family $\mathcal{R}$ consists of all
equivalence classes of separable unitary representations we immediately obtain
the following.

######  Corollary 1.6.

If the countable group $\Gamma$ has Kazhdan’s property (T) then
$\mathrm{Tr}\left(\Gamma\right)$ is a Bauer simplex.

This statement is inspired by the well-known result of Glasner and Weiss
[GW97] which deals with the geometry of the simplex of invariant probability
measures for a $\Gamma$-action on a compact space. We remark that Corollary
1.6 can specifically be deduced from the recent deep work of Kennedy and
Shamovich in which the authors develop a non-commutative Choquet theory and
extend the result of [GW97] to that realm [KS22, §14].

Note that there are non-Kazhdan groups whose simplex of traces is also Bauer.
For instance, if $\Gamma$ is discrete and abelian, then
$\mathrm{Ch}\left(\Gamma\right)$ coincides with its Pontryagin dual
$\widehat{\Gamma}$ which is compact. On the other hand, the simplex of traces
of many nilpotent groups, among them the discrete Heisenberg group, is not
Bauer [BV22, Example 1.9]. We do not know of any precise characterization of
the family of groups whose simplex of traces is Bauer.

The second ingredient of the proof of Theorem 1.1 deals with the geometry of
the subset of finite-dimensional traces.

######  Proposition 1.7.

Assume that the countable group $\Gamma$ has property
$\mathrm{T}(\mathcal{R})$. Then any finite-dimensional character
$\varphi\in\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$ is an
isolated point of the subset
$\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$.

Specializing once again to the case where the family $\mathcal{R}$ consists of
all equivalence classes of separable unitary representations, we immediately
obtain the following.

######  Corollary 1.8.

If the countable group $\Gamma$ has Kazhdan’s property $(\mathrm{T})$ then any
finite-dimensional character of $\Gamma$ is an isolated point of
$\mathrm{Ch}\left(\Gamma\right)$.

This is reminiscent of the definition of property (T) saying that finite-
dimensional representations are isolated in the Fell topology. In fact, it is
not hard to deduce Corollary 1.8 directly from that definition. The converse
direction of Corollary 1.8 is false however, as can be seen in some of the
examples following Theorem 1.1.

### On relative traces

Finally, we discuss a relative version of Theorem 1.6.

######  Theorem 1.9.

Let $\Lambda$ be a countable group admitting an action
$\theta:\Lambda\to\mathrm{Aut}\left(\Gamma\right)$ on the countable group
$\Gamma$. Assume that $\Lambda$ has Kazhdan’s property (T) and that
$\mathrm{Inn}(\Gamma)\leq\theta(\Lambda)$. Then

  * •

the space $\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$ consisting of all
$\Lambda$-invariant traces on the group $\Gamma$ is a Bauer simplex, and

  * •

finite-dimensional extremal $\Lambda$-invariant traces are isolated points of
the extreme point set
$\mathrm{Ch}_{\Lambda}\left(\Gamma\right)=\partial\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$.

To illustrate Theorem 1.9 consider the free abelian group
$\Gamma=\mathbb{Z}^{d}$ for some $d\geq 3$. Its group of automorphisms
$\Lambda=\mathrm{Aut}(\Gamma)\cong\mathrm{GL}_{d}(\mathbb{Z})$ has property
(T). It is an immediate consequence of Theorem 1.9 that any sequence of
pairwise distinct ergodic $\mathrm{GL}_{d}(\mathbb{Z})$-invariant measures on
the torus $\mathbb{T}^{d}$ converges to the Haar measure in the weak-$*$
topology, see Corollary 7.6. This fact is well known to experts for all $d\geq
2$. In any case, combining this information with vanishing results for
arithmetic groups (not necessarily semisimple ones, see Theorem 6.6) gives the
asymptotic vanishing of sequences of characters on
$\mathrm{GL}_{d}(\mathbb{Z})\ltimes\mathbb{Z}^{d}$ (Theorem 7.7).

Next, consider the free group $\Gamma=F_{d}$ of rank $d\geq 2$. Let
$\Lambda=\mathrm{Aut}\left(F_{d}\right)$ be its group of automorphisms. This
can be viewed as a non-commutative analogue of the discussion in the previous
paragraph. Unlike the simplex $\mathrm{Tr}\left(\mathbb{Z}^{d}\right)$, the
space of traces $\mathrm{Tr}\left(F_{d}\right)$ is as far as possible from
being a Bauer simplex — it is a Poulsen simplex, meaning that
$\overline{\mathrm{Ch}\left(F_{d}\right)}=\mathrm{Tr}\left(F_{d}\right)$
[OSV23]. However, the celebrated results of [KNO19, KKN21, Nit20] say that the
group of automorphisms $\mathrm{Aut}(F_{d})$ has property (T) provided $d\geq
4$.555The group $\mathrm{Aut}(F_{d})$ is not Kazhdan for $1\leq d\leq 3$. So
Theorem 1.9 can be brought to bear. Assuming $d\geq 4$ we conclude that the
space $\mathrm{Tr}_{\mathrm{Aut}(F_{d})}\left(F_{d}\right)$ is a Bauer simplex
after all.

$\operatorname{Aut}(F_{d})$-invariant positive definite functions on the free
group $F_{d}$ have recently attracted significant interest with connection to
word measures on groups and the Wiegold conjecture, see [CMP19] for a survey.
Such positive definite functions are in fact
$\operatorname{Out}(F_{d})$-invariant _traces_. The above result is an initial
step towards understanding their structure.

### Structure of the paper

In §2 and §3 we discuss general properties of traces and characters. In §4 we
study various notions of spectral gap of traces. These notions correspond to
closed conditions on the space of all traces. This analysis is used in §5 to
discuss the geometry of the simplex of traces on groups with property
$\mathrm{T}(\mathcal{R})$. In particular we tackle Theorem 1.5 and Proposition
1.7. Applications to character and representation limits are discussed in §6.
This is where our main results, Theorems 1.1, 1.3 and their generalizations
are proved. Lastly in §7 we discuss relative characters and prove Theorem 1.9.

### Acknowledgements

A.L was supported by ISF grant 1788/22 and BSF grant No. 2022105. R.S and I.V
were supported by ERC advanced grant No. 101052954.

##  2\. Traces and characters

We start with some preliminaries regarding traces and characters on groups.
For a comprehensive overview of this theory we refer to [BdlH20, Chapter 11].

Let $\Gamma$ be a countable discrete group. A _trace_ on $\Gamma$ is a
function $\varphi:\Gamma\to\mathbb{C}$ satisfying the following conditions:

  1. (1)

$\varphi$ is _positive definite_ :
$\sum_{i=1}^{n}{\alpha}_{i}\overline{\alpha}_{j}\varphi(\gamma_{j}^{-1}\gamma_{i})\geq
0$ for any $n\in\mathbb{N}$ and any choice of group elements
$\gamma_{1},...,\gamma_{n}\in\Gamma$ and scalars
$\alpha_{1},...,\alpha_{n}\in\mathbb{C}$.

  2. (2)

$\varphi$ is _conjugation-invariant_ : $\varphi(x^{-1}\gamma
x)=\varphi(\gamma)$ for all group elements $\gamma,x\in\Gamma$.

  3. (3)

$\varphi$ is _normalized_ : $\varphi(e_{\Gamma})=1$.

Let $\mathrm{Tr}\left(\Gamma\right)$ denote the space of all traces on the
group $\Gamma$ equipped with the topology of pointwise convergence. It is not
hard to see that $|\varphi(\gamma)|\leq\varphi(e_{\Gamma})=1$ holds for all
group elements $\gamma\in\Gamma$. As such, the set
$\mathrm{Tr}\left(\Gamma\right)$ can be regarded as a convex weak-$*$
compact666The weak-$*$ topology coincides with the topology of pointwise
convergence on $l^{\infty}(\Gamma)$. subset of $l^{\infty}(\Gamma)$. In
particular, any trace $\varphi$ on $\Gamma$ extends uniquely to a linear
functional on $\ell^{1}(\Gamma)$. We shall use this extension without further
mention.

Let $\mathrm{Ch}\left(\Gamma\right)$ denote the subset of extreme points of
the convex set $\mathrm{Tr}\left(\Gamma\right)$. Elements of
$\mathrm{Ch}\left(\Gamma\right)$ are called777Some authors refer to all traces
as “characters” and to extreme traces as “indecomposable characters”.
_characters_ of the group $\Gamma$.

Consider some trace $\varphi\in\mathrm{Tr}\left(\Gamma\right)$. Associated to
the trace $\varphi$ is the _GNS (Gelfand–Naimark–Segal) data_
$(\pi_{\varphi},\rho_{\varphi},\mathcal{H}_{\varphi},v_{\varphi})$. Here
$\mathcal{H}_{\varphi}$ is a Hilbert space and
$\pi_{\varphi},\rho_{\varphi}:\Gamma\to\mathrm{U}(\mathcal{H}_{\varphi})$ is a
pair of unitary representations. Moreover
$v_{\varphi}\in\mathcal{H}_{\varphi}$ is a cyclic unit vector for
$\pi_{\varphi}$ as well as for $\rho_{\varphi}$ satisfying
$\pi_{\varphi}(\gamma)v_{\varphi}=\rho_{\varphi}(\gamma^{-1})v_{\varphi}$ for
all elements $\gamma\in\Gamma$. The two representations $\pi_{\varphi}$ and
$\rho_{\varphi}$ commute, namely
$\pi_{\varphi}(\gamma_{1})\rho_{\varphi}(\gamma_{2})=\rho_{\varphi}(\gamma_{2})\pi_{\varphi}(\gamma_{1})$
for any pair of elements $\gamma_{1},\gamma_{2}\in\Gamma$. Lastly

(2.1)
$\varphi(\gamma)=\left<\pi_{\varphi}(\gamma)v_{\varphi},v_{\varphi}\right>=\left<\rho_{\varphi}(\gamma^{-1})v_{\varphi},v_{\varphi}\right>\quad\forall\gamma\in\Gamma.$

These requirements determine the GNS data uniquely up to an isomorphism. As
the group $\Gamma$ is countable the Hilbert space $\mathcal{H}_{\varphi}$ is
separable for every $\varphi\in\mathrm{Tr}\left(\Gamma\right)$.

Associated to the trace $\varphi\in\mathrm{Tr}\left(\Gamma\right)$ is the von
Neumann algebra $M_{\varphi}$ generated by the operators
$\pi_{\varphi}(\gamma)$ for all $\gamma\in\Gamma$. In other words
$M_{\varphi}=\pi_{\varphi}(\Gamma)^{\prime\prime}\subseteq\mathrm{B}(\mathcal{H}_{\varphi})$
by the von Neumann bicommutant theorem. The trace $\varphi$ is a character if
and only if the von Neumann algebra $M_{\varphi}$ is a _factor_ , namely
$\mathrm{Z}(M_{\varphi})=\mathbb{C}$. The group $\Gamma$ admits an _adjoint
representation_ on the von Neumann algebra $M_{\varphi}$ given by

(2.2)
$\mathrm{Ad}_{\gamma}(x)=\pi_{\varphi}(\gamma)x\pi_{\varphi}(\gamma^{-1})\quad\forall\gamma\in\Gamma,x\in
M_{\varphi}.$

### Dominated traces

Let $\varphi,\psi\in\mathrm{Tr}\left(\Gamma\right)$ be a pair of traces. We
say that the trace $\psi$ is _dominated_ by the trace $\varphi$ and write
$\psi\leq\varphi$ if there is some $0<\alpha\leq 1$ such that the function
$\varphi-\alpha\psi$ is positive definite. There is a surjective mapping from
the set of non-zero operators $T\in\mathrm{Z}(M_{\varphi})$ satisfying
$0<T\leq 1$ to the set of traces $\psi$ dominated by the trace $\varphi$. This
mapping takes such an element $T$ to the trace $\varphi_{T}$ given by

(2.3)
$\varphi_{T}(\gamma)=\frac{1}{\|T^{\frac{1}{2}}v_{\varphi}\|^{2}}\left<\pi_{\varphi}(\gamma)T^{\frac{1}{2}}v_{\varphi},T^{\frac{1}{2}}v_{\varphi}\right>\quad\forall\gamma\in\Gamma.$

Moreover $\varphi_{T_{1}}=\varphi_{T_{2}}$ for a pair of elements
$T_{1},T_{2}$ as above if and only if $T_{1}=\alpha T_{2}$ for some
$\alpha>0$. See [BdlH20, Lemma 11.C.2].

### The trivial character

The trivial character $1\in\mathrm{Ch}\left(\Gamma\right)$ is defined by
$1(\gamma)=1$ for all group elements $\gamma\in\Gamma$.

######  Proposition 2.1.

Let $\varphi\in\mathrm{Tr}\left(\Gamma\right)$ be a trace. The unitary
representation $\pi_{\varphi}$ has invariant vectors if and only if
$1\leq\varphi$.

###### Proof.

First, assume that the unitary representation $\pi_{\varphi}$ admits invariant
vectors. The orthogonal projection $T$ to the subspace of invariant vectors is
non-zero and satisfies $0\leq T\leq 1$ as well as
$T\in\mathrm{Z}(M_{\varphi})$. Therefore $\varphi_{T}\leq\varphi$ and
$\varphi_{T}=1$ by the formula in Equation (2.3). In the converse direction,
assume that $\varphi_{T}=1$ for some non-zero $T\in\mathrm{Z}(M_{\varphi})$
satisfying $0\leq T\leq 1$. Then
$T^{\frac{1}{2}}v_{\varphi}\in\mathcal{H}_{\varphi}$ is a non-zero invariant
vector. ∎

If $\varphi\in\mathrm{Ch}\left(\Gamma\right)$ and
$\psi\in\mathrm{Tr}\left(\Gamma\right)$ satisfy $\psi\leq\varphi$ then
certainly $\varphi=\psi$. Combined with Proposition 2.1 this gives the
following.

######  Corollary 2.2.

Let $\varphi\in\mathrm{Ch}\left(\Gamma\right)$ be a character. The unitary
representation $\pi_{\varphi}$ has invariant vectors if and only if $\varphi$
is the trivial character $1$.

Note that the statement of Corollary 2.2 is equivalent to saying that the
unitary representation $\pi_{\varphi}$ is trivial.

The above statements hold true in the same way if the unitary representation
$\pi_{\varphi}$ is replaced by the unitary representation $\rho_{\varphi}$.

### Finite-dimensional characters

Let $\mathcal{H}$ be a Hilbert space with conjugate Hilbert space
$\overline{\mathcal{H}}$. The Hilbertian tensor product
$\mathcal{H}\otimes\overline{\mathcal{H}}$ is naturally isomorphic to the
Hilbert space $\mathrm{HS}(\mathcal{H})$ of all Hilbert–Schmidt operators
acting on $\mathcal{H}$. The space $\mathrm{HS}(\mathcal{H})$ with its
Hilbert–Schmidt norm admits a dense subspace $\mathrm{FR}(\mathcal{H})$
consisting of all finite rank operators. Note that the identity map
$\mathrm{Id}$ belongs to $\mathrm{HS}(\mathcal{H})$ if and only if the Hilbert
space $\mathcal{H}$ is finite-dimensional.

Let $\pi$ be a unitary representation acting on the Hilbert space
$\mathcal{H}$. Then the unitary representation $\pi\otimes\pi^{*}$ acts on the
tensor product Hilbert space $\mathcal{H}\otimes\overline{\mathcal{H}}$, and
this action is equivalent to the conjugation action on
$\mathrm{HS}(\mathcal{H})$. An invariant vector $T\in\mathrm{HS}(\mathcal{H})$
for the unitary representation $\pi\otimes\pi^{*}$ is an operator satisfying
$\pi(\gamma)T=T\pi(\gamma)$ for all elements $\gamma\in\Gamma$. In other
words, such an invariant vector $T$ is an intertwining operator for the
representation $\pi$.

A trace $\varphi\in\mathrm{Tr}\left(\Gamma\right)$ is said to be _finite-
dimensional_ if its GNS unitary representation $\pi_{\varphi}$ is finite-
dimensional.

######  Lemma 2.3.

Let $\varphi\in\mathrm{Tr}\left(\Gamma\right)$ be a trace. The unitary
representation $\pi_{\varphi}\otimes\pi_{\varphi}^{*}$ has invariant vectors
if and only if $\psi\leq\varphi$ for some finite-dimensional character
$\psi\in\mathrm{Ch}\left(\Gamma\right)$.

###### Proof.

Assume that there is a finite-dimensional character
$\psi\in\mathrm{Ch}\left(\Gamma\right)$ given by $\psi=\varphi_{T}$ for some
non-zero operator $T\in\mathrm{Z}(M_{\varphi})$ satisfying $0\leq T\leq 1$.
Consider the closed subspace $\mathcal{L}$ of the Hilbert space
$\mathcal{H}_{\varphi}$ given by

(2.4)
$\mathcal{L}=\overline{\mathrm{span}_{\mathbb{C}}\\{\pi_{\varphi}(\gamma)v_{\psi}\>:\>\gamma\in\Gamma\\}}$

where $v_{\psi}=T^{\frac{1}{2}}v_{\varphi}$. The GNS data associated to the
character $\psi$ is uniquely determined up to an isomorphism. This implies
that the subspace $\mathcal{L}$ is isomorphic to the Hilbert space
$\mathcal{H}_{\psi}$. As such, the closed subspace $\mathcal{L}$ is finite-
dimensional. Therefore the orthogonal projection operator $\mathrm{P}$ to the
subspace $\mathcal{L}$ belongs to $\mathrm{HS}(\mathcal{H}_{\varphi})$.
Moreover $\mathrm{P}$ is an invariant vector for the unitary representation
$\pi_{\varphi}\otimes\pi_{\varphi}^{*}$ as the subspace $\mathcal{L}$ is
$\pi_{\varphi}(\Gamma)$-invariant.

Conversely, suppose that the unitary representation
$\pi_{\varphi}\otimes\pi^{*}_{\varphi}$ admits invariant vectors. It follows
that the unitary representation $\pi_{\varphi}$ admits a finite-dimensional
sub-representation $\pi_{0}$ [BdLHV08, A.1.12]. Let $\mathrm{P}$ be the
orthogonal projection to the sum of all the sub-representations of
$\pi_{\varphi}$ isomorphic to $\pi_{0}$. Then
$\mathrm{P}\in\pi_{\varphi}(\Gamma)^{\prime}$. In addition
$\mathrm{P}\in\pi_{\varphi}(\Gamma)^{\prime\prime}$ as the von Neumann algebra
generated by a direct sum of disjoint representations is the direct sum of the
corresponding von Neumann algebras [BdlH20, Proposition 6.B.4]. It follows
that $\mathrm{P}\in\mathrm{Z}(M_{\varphi})$. Denote
$\psi=\varphi_{\mathrm{P}}$ so that $\psi\leq\varphi$. Note that the trace
$\psi$ is finite-dimensional. ∎

Restricting once again to the case where $\varphi$ is a character, we get the
following useful characterization.

######  Corollary 2.4.

Let $\varphi\in\mathrm{Ch}\left(\Gamma\right)$ be a character. Then $\varphi$
is finite-dimensional if and only if the representation
$\pi_{\varphi}\otimes\pi^{*}_{\varphi}$ has invariant vectors.

### Amenable traces

A trace $\varphi\in\mathrm{Tr}\left(\Gamma\right)$ is called _amenable_ if the
corresponding unitary representation $\pi_{\varphi}$ is amenable in the sense
of Bekka [Bek90], namely the tensor product representation
$\pi_{\varphi}\otimes\pi^{*}_{\varphi}$ admits almost-invariant vectors. This
is equivalent to saying that the norm of the positive contracting operator
$(\pi_{\varphi}\otimes\pi_{\varphi}^{*})(a)$ is $1$ for some choice of an
element $a\in\ell^{1}(\Gamma)$ such that the support of the element $a^{*}a$
generates the group $\Gamma$ [BdLHV08, G.4.2]. A non-amenable trace cannot
dominate any finite-dimensional trace, see Lemma 2.3. In particular, a non-
amenable character is infinite-dimensional.

##  3\. The conjugation representation

In this section, we turn our attention to the conjugation representation
associated to a trace. This representation will be helpful in determining
whether the trace in question is in fact a character.

Let $\Gamma$ be a countable discrete group and let
$\varphi\in\mathrm{Tr}\left(\Gamma\right)$ be a trace on $\Gamma$. Consider
the unitary representation
$c_{\varphi}:\Gamma\to\mathrm{U}(\mathcal{H}_{\varphi})$ given by

(3.1)
$c_{\varphi}(\gamma)=\pi_{\varphi}(\gamma)\rho_{\varphi}(\gamma)=\rho_{\varphi}(\gamma)\pi_{\varphi}(\gamma)\quad\forall\gamma\in\Gamma.$

Note that the representation $c_{\varphi}$ is the restriction of the
representation

(3.2)
$\pi_{\varphi}\times\rho_{\varphi}:\Gamma\times\Gamma\to\mathrm{U}(\mathcal{H}_{\varphi})$

to the diagonal subgroup $\Gamma\cong\Delta\Gamma\leq\Gamma\times\Gamma$.

Let $M_{\varphi}$ be the von Neumann algebra associated to the trace
$\varphi$. Consider the linear map

(3.3) $\iota_{\varphi}:M_{\varphi}\to\mathcal{H}_{\varphi},\quad\iota:x\mapsto
xv_{\varphi}\quad\forall x\in M_{\varphi}.$

The fact that the vector $v_{\varphi}$ is cyclic for the representation
$\pi_{\varphi}$ implies that the image of $\iota_{\varphi}$ is dense.
Likewise, the fact that the vector $v_{\varphi}$ is cyclic for the
representation $\rho_{\varphi}$ combined with the fact that
$\rho_{\varphi}(\Gamma)\subset M^{\prime}_{\varphi}$ implies that
$\iota_{\varphi}$ is injective. The map $\iota_{\varphi}$ intertwines the
adjoint representation of the group $\Gamma$ on the algebra $M_{\varphi}$ with
the representation $c_{\varphi}$ on the Hilbert space $\mathcal{H}_{\varphi}$,
namely

(3.4)
$\displaystyle\begin{split}\iota_{\varphi}(\mathrm{Ad}_{\gamma}(x))&=\mathrm{Ad}_{\gamma}(x)v_{\varphi}=\left(\pi_{\varphi}(\gamma)x\pi_{\varphi}(\gamma^{-1})\right)v_{\varphi}=\pi_{\varphi}(\gamma)x\rho_{\varphi}(\gamma)v_{\varphi}=\\\
&=\pi_{\varphi}(\gamma)\rho_{\varphi}(\gamma)xv_{\varphi}=c_{\varphi}(\gamma)xv_{\varphi}=c_{\varphi}(\gamma)\iota_{\varphi}(x)\end{split}$

for all group elements $\gamma\in\Gamma$ and all $x\in M_{\varphi}$. If
follows that the image of $\iota_{\varphi}$ is a $c_{\varphi}$-invariant
subspace of $\mathcal{H}_{\varphi}$ and that the center
$\mathrm{Z}(M_{\varphi})$ is mapped injectively into the subspace of
$c_{\varphi}$-invariant vectors $\mathcal{H}_{\varphi}^{c_{\varphi}}$.

The main goal of the current section is the following result providing
information about the image of the linear map $\iota_{\varphi}$.

######  Proposition 3.1.

The embedding
$\iota_{\varphi}:\mathrm{Z}(M_{\varphi})\hookrightarrow\mathcal{H}_{\varphi}^{c_{\varphi}}$
has dense image.

The proof of Proposition 3.1 will be given below. As was already mentioned,
the trace $\varphi$ is a character if and only if the center
$\mathrm{Z}(M_{\varphi})$ of the von Neumann algebra $M_{\varphi}$ is one-
dimensional. Therefore Proposition 3.1 immediately implies the following fact.
While this is known to experts, we are not aware of any straightforward
reference.

######  Corollary 3.2.

The trace $\varphi$ is a character if and only if
$\dim_{\mathbb{C}}\mathcal{H}_{\varphi}^{c_{\varphi}}=1$.

The condition $\dim_{\mathbb{C}}\mathcal{H}_{\varphi}^{c_{\varphi}}=1$ is
equivalent to saying that
$\mathcal{H}_{\varphi}^{c_{\varphi}}=\mathrm{span}_{\mathbb{C}}v_{\varphi}$.
We generalize this statement to the relative setting in §7.

### Dense linear subspaces of Hilbert spaces

The proof of Proposition 3.1 will rely on the following general lemma
concerning Hilbert spaces.

######  Lemma 3.3.

Let $V$ be a Hilbert space. Let $C$ be a closed convex balanced888Recall that
a subset $C$ of a complex vector space is called _balanced_ if $aC\subset C$
for all scalars $a$ with $|a|\leq 1$. subset of $V$ such that the subspace
$A=\bigcup_{n}nC$ is dense in $V$. Denote by $\Pi$ the nearest point
projection from $V$ to $C$. Then $A\cap W$ is dense in $W$ for any closed
subspace $W\leq V$ satisfying $\Pi(W)\subset W$.

###### Proof.

Let $W$ be any closed subspace of $V$ satisfying $\Pi(W)\subset W$. Consider
any vector $v\in W\cap(A\cap W)^{\perp}$. We will show that necessarily $v=0$.

We start by showing that $\Pi(v)=0$. Note that the nearest point projection
$\Pi(v)$ satisfies

(3.5) $\Pi(v)\in C\cap W\subset A\cap W.$

The properties of the nearest point projection $\Pi(v)$ imply that for any
other vector $w\in C$ the angle at the point $\Pi(v)$ between the two line
segments $\left[\Pi(v),v\right]$ and $\left[\Pi(v),w\right]$ cannot be acute.
In terms of the inner product this means that

(3.6) $\Re\left<v-\Pi(v),w-\Pi(v)\right>\leq 0\quad\forall w\in C.$

Substituting $w=0$ and recalling that $v\in(A\cap W)^{\perp}$ while $\Pi(v)\in
A\cap W$ gives $\Re\left<\Pi(v),\Pi(v)\right>\leq 0$. Therefore $\Pi(v)=0$. As
Hilbert spaces are strictly convex, we conclude that

(3.7) $\|v\|=\|v-\Pi(v)\|<\|v-w\|$

for all vectors $w\in C$ with $w\neq\Pi(v)=0$.

The linear subspace $A$ is dense in the Hilbert space $V$ by assumption.
Therefore there is a sequence of vectors $u_{n}\in nC$ converging to $v$.
Denote $x_{n}=\frac{1}{n}u_{n}$ so that $x_{n}\in C$ for all $n\in\mathbb{N}$
and $x_{n}\to 0$. As $u_{n}\to v$ we have that
$\left<u_{n},v\right>\to\left<v,v\right>$ as well as
$\left<u_{n},u_{n}\right>\to\left<v,v\right>$. Assume towards contradiction
that $v\neq 0$. In that case

(3.8)
$\frac{2}{n}\Re\left<u_{n},v\right>>\frac{1}{n^{2}}\left<u_{n},u_{n}\right>$

for all $n$ sufficiently large. On the other hand, for all $n$ we have that

(3.9)
$\left<v-x_{n},v-x_{n}\right>=\left<v,v\right>-\frac{2}{n}\Re\left<u_{n},v\right>+\frac{1}{n^{2}}\left<u_{n},u_{n}\right>.$

Putting together the two Equations (3.8) and (3.9) implies that
$\|v\|>\|v-x_{n}\|$ for all $n$ sufficiently large. It follows from the
discussion involving Equation (3.7) that $x_{n}=u_{n}=0$ for all $n$
sufficiently large. As $u_{n}\to v$ we arrive at a contradiction to the
assumption $v\neq 0$ and deduce that $v=0$ as required. ∎

### Invariant vectors for the conjugation representation

Let $(M_{\varphi})_{1}$ denote the subset of the von Neumann algebra
$M_{\varphi}\subset\mathrm{B}(\mathcal{H}_{\varphi})$ consisting of all
operators with operator norm at most $1$. Identify the von Neumann algebra
$M_{\varphi}$ as a subspace of the Hilbert space $\mathcal{H}_{\varphi}$ via
the map $\iota_{\varphi}$ introduced in Equation (3.3). This identification
makes $(M_{\varphi})_{1}$ a closed subset of the Hilbert space
$\mathcal{H}_{\varphi}$ by [Pet16, Corollary 5.2]. Moreover the subset
$(M_{\varphi})_{1}$ is convex and $c_{\varphi}$-invariant. We are ready to
complete the following proof.

###### Proof of Proposition 3.1.

The desired statement follows immediately as a special case of Lemma 3.3. We
let $V$ be the Hilbert space $\mathcal{H}_{\varphi}$, $W$ be the Hilbert
subspace of $c_{\varphi}$-invariant vectors and $C$ be the closed convex
subset $(M_{\varphi})_{1}$ of $\mathcal{H}_{\varphi}$. In this situation $\Pi$
denotes the nearest point projection from $\mathcal{H}_{\varphi}$ to the
closed convex subset $(M_{\varphi})_{1}$. As $c_{\varphi}$ acts by isometries,
the subset $(M_{\varphi})_{1}$ is $c_{\varphi}$-invariant and the projection
$\Pi$ is $c_{\varphi}$-equivariant. Therefore $\Pi$ preserves the subspace $W$
of all $c_{\varphi}$-invariant vectors, as is required for the purpose of
Lemma 3.3. ∎

##  4\. Traces and spectral gap

Let $\Gamma$ be a countable discrete group and
$\varphi\in\mathrm{Tr}\left(\Gamma\right)$ be a fixed trace with associated
GNS data $(\pi_{\varphi},\rho_{\varphi},\mathcal{H}_{\varphi},v_{\varphi})$.
In addition, consider the unitary conjugation representation
$c_{\varphi}:\Gamma\to\mathrm{U}(\mathcal{H}_{\varphi})$ introduced in §3. In
this key section, we show that spectral gap for any of the representations
$\pi_{\varphi},\pi_{\varphi}\otimes\pi^{*}_{\varphi}$ and $c_{\varphi}$ can be
expressed in terms of inequalities involving elements of the group algebra
$\mathbb{C}\left[\Gamma\right]$. The exact formulas will not be crucial.
Rather, their existence demonstrates that spectral gap associated to traces is
preserved when taking limits. This concept resonates with Kesten’s theorem,
though our analysis demands a more detailed approach.

Fix an arbitrary element $a\in\ell^{1}(\Gamma)$. Typically the element $a$
will be chosen so that its support generates the group $\Gamma$, but this is
not necessary at this point. Set $b=a^{*}a\in\ell^{1}(\Gamma)$ and denote

(4.1) $b=\sum_{\gamma\in\Gamma}b_{\gamma}\gamma$

with the coefficients $b_{\gamma}\in\mathbb{C}$ for each $\gamma\in\Gamma$. We
will use these notations throughout this section without further mention.

In what follows we will repeatedly make use of the formula

(4.2)
$\|\pi_{\varphi}(x)v_{\varphi}\|^{2}=\left<\pi_{\varphi}(x)v_{\varphi},\pi_{\varphi}(x)v_{\varphi}\right>=\left<\pi_{\varphi}(x^{*}x)v_{\varphi},v_{\varphi}\right>=\varphi(x^{*}x)$

holding true for all elements $x\in\ell^{1}(\Gamma)$.

######  Lemma 4.1.

Fix $\beta\geq 0$. Then $\|\pi_{\varphi}(a)\|\leq\beta^{\frac{1}{2}}$ if and
only if

(4.3) $\varphi(x^{*}(b-\beta)x)\leq 0$

for every element $x\in\mathbb{C}\left[\Gamma\right]$.

###### Proof.

Since the collection of vectors $\pi_{\varphi}(x)v_{\varphi}$ with
$x\in\mathbb{C}\left[\Gamma\right]$ is dense in the Hilbert space
$\mathcal{H}_{\varphi}$, it suffices to estimate the norm of the operator
$\pi_{\varphi}(a)$ on this subset. In particular, the estimate
$\|\pi_{\varphi}(a)\|\leq\beta^{\frac{1}{2}}$ is equivalent to the inequality

(4.4)
$\|\pi_{\varphi}(a)\pi_{\varphi}(x)v_{\varphi}\|\leq\beta^{\frac{1}{2}}\|\pi_{\varphi}(x)v_{\varphi}\|$

holding true for every $x\in\mathbb{C}\left[\Gamma\right]$. The formula in
Equation (4.2) shows that Equation (4.4) is equivalent for each individual
element $x\in\mathbb{C}\left[\Gamma\right]$ to the inequality

(4.5) $\varphi(x^{*}bx)=\varphi((ax)^{*}(ax))\leq\beta\varphi(x^{*}x).$

The desired formula in Equation (4.3) follows by rearranging Equation (4.5). ∎

The next estimate is concerned with the conjugation representation
$c_{\varphi}$.

######  Lemma 4.2.

Fix $\beta\geq 0$. Then

(4.6)
$\|c_{\varphi}(a)_{|\mathrm{span}_{\mathbb{C}}(v_{\varphi})^{\perp}}\|\leq\beta^{\frac{1}{2}}$

if and only if the inequality

(4.7) $\varphi\left(\sum_{\gamma}b_{\gamma}\gamma^{*}x^{*}\gamma x-\beta
x^{*}x\right)\leq(1-\beta)|\varphi(x)|^{2}$

holds true for every element $x\in\mathbb{C}\left[\Gamma\right]$

###### Proof.

The collection of vectors of the form $\pi_{\varphi}(x)v_{\varphi}$ for some
element $x\in\mathbb{C}\left[\Gamma\right]$ forms a dense subset of the
Hilbert space $\mathcal{H}_{\varphi}$. Therefore, to estimate the norm of the
operator $c_{\varphi}(a)$ it suffices to restrict attention to this subset.
The operator norm bound
$\|c_{\varphi}(a)_{|\mathrm{span}_{\mathbb{C}}(v_{\varphi})^{\perp}}\|\leq\beta^{\frac{1}{2}}$
is equivalent to the inequality

(4.8)
$\beta\|w\|^{2}-\|c_{\varphi}(a)w\|^{2}\geq(\beta-1)|\left<w,v_{\varphi}\right>|^{2}$

for all vectors of the form $w=\pi_{\varphi}(x)v_{\varphi}$ for some
$x\in\mathbb{C}\left[\Gamma\right]$, see the auxiliary Lemma 4.5. Let us
estimate each one of the three terms appearing in Equation (4.8). First

(4.9) $\|w\|^{2}=\|\pi_{\varphi}(x)v_{\varphi}\|^{2}=\varphi(x^{*}x)$

according to Equation (4.2). Next we compute $\|c_{\varphi}(a)w\|^{2}$.
Observe that

(4.10) $\displaystyle\begin{split}\left\langle
c_{\varphi}(\gamma)w,w\right\rangle&=\left\langle
c_{\varphi}(\gamma)\pi_{\varphi}(x)v_{\varphi},\pi_{\varphi}(x)v_{\varphi}\right\rangle=\left\langle\pi_{\varphi}(\gamma)\rho_{\varphi}(\gamma)\pi_{\varphi}(x)v_{\varphi},\pi_{\varphi}(x)v_{\varphi}\right\rangle=\\\
&=\left\langle\pi_{\varphi}(x^{*})\pi_{\varphi}(\gamma)\pi_{\varphi}(x)\rho_{\varphi}(\gamma)v_{\varphi},v_{\varphi}\right\rangle=\varphi\left(\gamma^{*}x^{*}\gamma
x\right)\end{split}$

for all elements $\gamma\in\Gamma$. We have used the formula
$\rho_{\varphi}(z)v_{\varphi}=\pi_{\varphi}(z^{*})v_{\varphi}$ as well as the
trace property $\tau(zz^{\prime})=\tau(z^{\prime}z)$ which hold true for all
elements $z,z^{\prime}\in M_{\varphi}$. Therefore we get

(4.11) $\|c_{\varphi}(a)w\|^{2}=\left\langle
c_{\varphi}(b)w,w\right\rangle=\sum_{\gamma\in\Gamma}\left\langle
b_{\gamma}c_{\varphi}(\gamma)w,w\right\rangle=\varphi\left(\sum_{\gamma\in\Gamma}b_{\gamma}\gamma^{*}x^{*}\gamma
x\right).$

Lastly the inner product $\left<w,v_{\varphi}\right>$ is simply given by

(4.12)
$\left<w,v_{\varphi}\right>=\left<\pi_{\varphi}(x)v_{\varphi},v_{\varphi}\right>=\varphi(x).$

The desired formula in Equation (4.7) follows by combining these three
computations. ∎

The following two results are extensions of Lemma 4.1 to the tensor product
representation $\pi_{\varphi}\otimes\pi^{*}_{\varphi}$. We identify
$\pi_{\varphi}^{*}$ with the unitary representation associated to the
conjugate trace $\overline{\varphi}\in\mathrm{Tr}\left(\Gamma\right)$
admitting the GNS data
$(\pi_{\varphi}^{*},\rho_{\varphi}^{*},\mathcal{H}^{*}_{\varphi},v^{*}_{\varphi})$.

######  Lemma 4.3.

Assume that the Hilbert space $\mathcal{H}_{\varphi}$ is infinite-dimensional.
Fix $\beta\geq 0$. Then

(4.13) $\|(\pi_{\varphi}\otimes\pi_{\varphi}^{*})(a)\|\leq\beta^{\frac{1}{2}}$

if and only if

(4.14)
$\sum_{i,j=1}^{k}\left(\sum_{\gamma\in\Gamma}b_{\gamma}\varphi(x_{j}^{*}\gamma
x_{i})\overline{\varphi}\left(y_{j}^{*}\gamma
y_{i}\right)-\beta\varphi(x_{j}^{*}x_{i})\overline{\varphi}(y_{j}^{*}y_{i})\right)\leq
0$

for every $k\in\mathbb{N}$ and every
$x_{1},\ldots,x_{k}\in\mathbb{C}\left[\Gamma\right]$ and
$y_{1},\ldots,y_{k}\in\mathbb{C}\left[\Gamma\right]$.

###### Proof.

The collection of all the vectors of the form

(4.15)
$w=w^{k}_{x_{i},y_{i}}=\sum_{i=1}^{k}\pi_{\varphi}(x_{i})v_{\varphi}\otimes\pi^{*}_{\varphi}(y_{i})v^{*}_{\varphi}$

for some $k\in\mathbb{N}$ and some
$x_{i},y_{i}\in\mathbb{C}\left[\Gamma\right]$ is a dense subset of the Hilbert
space
$\mathcal{H}_{\varphi}\otimes\mathcal{H}_{\varphi}^{*}\cong\mathrm{HS}(\mathcal{H}_{\varphi})$.
Therefore, the operator norm bound
$\|(\pi_{\varphi}\otimes\pi_{\varphi}^{*})(a)\|^{2}\leq\beta$ is equivalent to
the inequality

(4.16) $\|(\pi_{\varphi}\otimes\pi_{\varphi}^{*})(a)w\|^{2}\leq\beta\|w\|^{2}$

holding true for every vector $w=w^{k}_{x_{i},y_{i}}$ as in Equation (4.15)
above. It is possible to rewrite the norm of the vector $w$ in terms of the
trace $\varphi$, namely

(4.17) $\displaystyle\begin{split}\|w\|^{2}=\langle
w,w\rangle&=\sum_{i,j=1}^{k}\left<\pi_{\varphi}(x_{i})v_{\varphi}\otimes\pi^{*}_{\varphi}(y_{i})v^{*}_{\varphi},\pi_{\varphi}(x_{j})v_{\varphi}\otimes\pi^{*}_{\varphi}(y_{j})v^{*}_{\varphi}\right>\\\
&=\sum_{i,j=1}^{k}\left<\pi_{\varphi}(x_{i})v_{\varphi},\pi_{\varphi}(x_{j})v_{\varphi}\right>\cdot\left<\pi^{*}_{\varphi}(y_{i})v^{*}_{\varphi},\pi^{*}_{\varphi}(y_{j})v^{*}_{\varphi}\right>\\\
&=\sum_{i,j=1}^{k}\varphi(x_{j}^{*}x_{i})\overline{\varphi}(y_{j}^{*}y_{i}).\end{split}$

The next step is to estimate the norm
$\|(\pi_{\varphi}\otimes\pi_{\varphi}^{*})(a)w\|^{2}$. First, note that for
each element $\gamma\in\Gamma$ individually we have

(4.18)
$\displaystyle\begin{split}\left\langle\pi_{\varphi}\otimes\pi_{\varphi}^{*}(\gamma)w,w\right\rangle&=\sum_{i,j=1}^{k}\left\langle\pi_{\varphi}(\gamma)\pi_{\varphi}(x_{i})v_{\varphi},\pi_{\varphi}(x_{j})v_{\varphi}\right\rangle\cdot\left\langle\pi_{\varphi}^{*}(\gamma)\pi_{\varphi}^{*}\left(y_{i}\right)v_{\varphi}^{*},\pi_{\varphi}^{*}(y_{j})v_{\varphi}^{*}\right\rangle\\\
&=\sum_{i,j=1}^{k}\left\langle\pi_{\varphi}(x_{j}^{*}\gamma
x_{i})v_{\varphi},v_{\varphi}\right\rangle\cdot\left\langle\pi_{\varphi}^{*}\left(y_{j}^{*}\gamma
y_{i}\right)v_{\varphi}^{*},v_{\varphi}^{*}\right\rangle\\\
&=\sum_{i,j=1}^{k}\varphi(x_{j}^{*}\gamma
x_{i})\bar{\varphi}\left(y_{j}^{*}\gamma y_{i}\right).\end{split}$

Using the fact that $b=a^{*}a$ and summing Equation (4.18) over all $\gamma$’s
we get

(4.19)
$\displaystyle\begin{split}\|\pi_{\varphi}\otimes\pi_{\varphi}^{*}(a)w\|^{2}=\left\langle\pi_{\varphi}\otimes\pi_{\varphi}^{*}(b)w,w\right\rangle&=\sum_{\gamma\in\Gamma}b_{\gamma}\left\langle\pi_{\varphi}\otimes\pi_{\varphi}^{*}(\gamma)w,w\right\rangle\\\
&=\sum_{i,j=1}^{k}\sum_{\gamma\in\Gamma}b_{\gamma}\varphi(x_{j}^{*}\gamma
x_{i})\overline{\varphi}\left(y_{j}^{*}\gamma y_{i}\right).\end{split}$

The desired Equation (4.16) follows from the two Equations (4.18) and (4.19)
by rearranging the terms. ∎

We now consider the situation when unitary representation $\pi_{\varphi}$ is
finite-dimensional. In this case, we only obtain a one-way implication.

######  Lemma 4.4.

Assume that the Hilbert space $\mathcal{H}_{\varphi}$ is finite-dimensional so
that $\mathrm{Id}\in\mathrm{HS}(\mathcal{H}_{\varphi})$. Fix $\beta\geq 0$. If

(4.20)
$\|(\pi_{\varphi}\otimes\pi_{\varphi}^{*})(a)_{|\mathrm{span}_{\mathbb{C}}(\mathrm{Id})^{\perp}}\|\leq\beta^{\frac{1}{2}}$

then

(4.21)
$\sum_{i,j=1}^{k}\sum_{\gamma\in\Gamma}b_{\gamma}\varphi(x_{j}^{*}\gamma
x_{i})\overline{\varphi}\left(y_{j}^{*}\gamma
y_{i}\right)-\beta\varphi(x_{j}^{*}x_{i})\overline{\varphi}(y_{j}^{*}y_{i})\leq(1-\beta)\frac{\left(\sum_{i=1}^{k}\|x_{i}\|_{1}\|y_{i}\|_{1}\right)^{2}}{(\dim_{\mathbb{C}}\mathcal{H}_{\varphi})^{3}}$

for every $k\in\mathbb{N}$ and every
$x_{1},\ldots,x_{k}\in\mathbb{C}\left[\Gamma\right]$ and
$y_{1},\ldots,y_{k}\in\mathbb{C}\left[\Gamma\right]$.

###### Proof.

The operator norm bound in Equation (4.20) is equivalent to the inequality

(4.22)
$\|(\pi_{\varphi}\otimes\pi_{\varphi}^{*})(a)w\|^{2}-\beta\|w\|^{2}\leq\frac{(1-\beta)}{\dim_{\mathbb{C}}\mathcal{H}_{\varphi}}|\left<w,\mathrm{Id}\right>|^{2}$

holding true for any vector $w\in\mathrm{HS}(\mathcal{H}_{\varphi})$ of the
form defined in Equation (4.15), see the auxiliary Lemma 4.5 below. We have
obtained formulas for the norms $\|w\|^{2}$ as well as
$\|(\pi_{\varphi}\otimes\pi^{*}_{\varphi})(a)w\|^{2}$ in terms of the trace
$\varphi$, see Equations (4.17) and (4.18).

To conclude the proof it remains to estimate the inner product
$\left<w,\mathrm{Id}\right>$. Recall that the inner product on the
Hilbert–Schmidt space $\mathrm{HS}(\mathcal{H}_{\varphi})$ is given explicitly
by

(4.23) $\langle
T,S\rangle=\frac{1}{\dim_{\mathbb{C}}\mathcal{H}_{\varphi}}\mathrm{tr}(S^{*}T)$

for each pair of operators $T,S\in\mathrm{HS}(\mathcal{H}_{\varphi})$. In
particular $\|\mathrm{Id}\|^{2}=\dim_{\mathbb{C}}\mathcal{H}_{\varphi}$.

First consider the special case $k=1,x=\delta_{\gamma_{1}}$ and
$y=\delta_{\gamma_{2}}$ where $\gamma_{1},\gamma_{2}\in\Gamma$ are arbitrary
elements so that
$w=\pi_{\varphi}(\gamma_{1})v_{\varphi}\otimes\pi^{*}_{\varphi}(\gamma_{2})v^{*}_{\varphi}$
has rank $1$ as an operator on $\mathcal{H}_{\varphi}$. In that case

(4.24)
$|\left<w,\mathrm{Id}\right>|=|\left<\pi_{\varphi}(\gamma_{1})v_{\varphi}\otimes\pi^{*}_{\varphi}(\gamma_{2})v^{*}_{\varphi},\mathrm{Id}\right>|\leq\frac{1}{\dim_{\mathbb{C}}\mathcal{H}_{\varphi}}.$

In the general case, relying on the triangle inequality we get

(4.25)
$|\left<w,\mathrm{Id}\right>|\leq\frac{\sum_{i=1}^{k}\|x_{i}\|_{1}\|y_{i}\|_{1}}{\dim_{\mathbb{C}}\mathcal{H}_{\varphi}}.$

Substituting this Equation (4.25) in the previous Equation (4.22), we get the
desired inequality in Equation (4.21). ∎

The following auxiliary computation was used twice in the above proofs.

######  Lemma 4.5.

Let $A$ be a bounded linear operator acting on the Hilbert space
$\mathcal{H}$. Let $v\in\mathcal{H}$ be an $A$-invariant _unit_ vector. Fix
some $0<\beta<1$. Then

(4.26) $\|A_{|\mathrm{span}_{\mathbb{C}}(v)^{\perp}}\|\leq\beta^{\frac{1}{2}}$

if and only if

(4.27) $\beta\|w\|^{2}-\|Aw\|^{2}\geq(\beta-1)|\left<w,v\right>|^{2}$

for all vectors $w\in\mathcal{H}$.

###### Proof.

Denote $\mathcal{H}_{0}=\mathrm{span}_{\mathbb{C}}\\{v\\}$. Consider the
$A$-invariant direct sum decomposition
$\mathcal{H}=\mathcal{H}_{0}\oplus\mathcal{H}_{1}$ where
$\mathcal{H}_{1}=\mathcal{H}_{0}^{\perp}$. Any vector $w\in\mathcal{H}$ can be
written uniquely as $w=w_{0}+w_{1}$ with
$w_{0}=\left<w,v\right>v\in\mathcal{H}_{0}$ and
$w_{1}=w-w_{0}\in\mathcal{H}_{1}$.

To begin with, assume that Equation (4.27) holds for all vectors
$w\in\mathcal{H}$. Note that every vector $w\in\mathcal{H}_{1}$ satisfies
$\left<w,v\right>=0$. In particular Equation (4.27) reads
$\|Aw\|^{2}\leq\beta\|w\|^{2}$ with respect to any such vector, as required.

Conversely, assume that $\|A_{|\mathcal{H}_{1}}\|\leq\beta^{\frac{1}{2}}$.
Therefore any vector $w\in\mathcal{H}$ satisfies

(4.28)
$\|Aw\|^{2}=\|Aw_{0}\|^{2}+\|Aw_{1}\|^{2}\leq\|w_{0}\|^{2}+\beta\|w_{1}\|^{2}=(1-\beta)\|w_{0}\|^{2}+\beta\|w\|^{2}.$

The desired Equation (4.27) follows from Equation (4.28) noting that
$\|w_{0}\|=|\langle w,v\rangle|$. ∎

##  5\. Spectral gap and convergence of traces

Let $\Gamma$ be a countable discrete group. We study the behaviour of various
spectral gap properties with respect to taking limits of sequences of traces
on the group $\Gamma$.

Let $\mathcal{R}$ be a set of equivalence classes of unitary representations
of $\Gamma$. For each unitary representation $\pi\in\mathcal{R}$ let $\pi_{0}$
denote its sub-representation obtained by taking the orthogonal complement to
the subspace of invariant vectors.

######  Definition.

The group $\Gamma$ has _property $\mathrm{T}(\mathcal{R})$_ if the trivial
representation $1\in\widehat{\Gamma}$ is an isolated point of the set

(5.1) $\\{\pi_{0}\>:\>\pi\in\mathcal{R}\\}\cup\\{1\\}$

with respect to the Fell topology.

Property $\mathrm{T}(\mathcal{R})$ for the group $\Gamma$ is equivalent to the
following condition:

  * •

If $\rho$ is a unitary representation with $1\leq\rho$ then
$\rho\notin\overline{\\{\pi_{0}\>:\>\pi\in\mathcal{R}\\}}$ with respect to the
Fell topology.

In the special case where every representation $\pi\in\mathcal{R}$ is
irreducible, i.e. the set of representations $\mathcal{R}$ is contained in the
unitary dual $\widehat{\Gamma}$, the above definition of property
$\mathrm{T}(\mathcal{R})$ coincides with the one given by Lubotzky and Zimmer
in [LZ89, Definition 1.1]. For our purposes it will be more convenient to
allow for reducible representations as well.

Here are some well known examples of property $\mathrm{T}(\mathcal{R})$.

  * •

Let $\mathrm{Rep}_{\aleph_{0}}(\Gamma)$ be the set of equivalence classes of
all separable unitary representations of the countable group $\Gamma$.
Property $\mathrm{T}(\mathrm{Rep}_{\aleph_{0}}(\Gamma))$ is the same as
Kazhdan’s property (T).

  * •

Let $\mathrm{FD}$ be the family of all equivalence classes of finite-
dimensional unitary representations of the group $\Gamma$. Property
$\mathrm{T}(\mathrm{FD})$ has been considered in the literature [LZ89].

  * •

Property $(\tau)$ can be regarded as a special case of property
$\mathrm{T}(\mathcal{R})$ for a particular choice of the set $\mathcal{R}$.
Indeed, given a family $\mathcal{N}$ of normal subgroups of the group $\Gamma$
we denote by $\mathcal{R}(\mathcal{N})$ the set of all unitary representation
factoring through some normal subgroup in $\mathcal{N}$. Property $(\tau)$ is
the same as property $\mathrm{T}(\mathcal{R}(\mathcal{F}))$ where
$\mathcal{F}$ is the family of all finite index normal subgroups of the group
$\Gamma$. See [LZ05] for more information.

  * •

In the special case where $\Gamma$ is an arithmetic group, _Selberg’s
property_ is the same as property $\mathrm{T}(\mathcal{R}(\mathcal{C}))$ where
$\mathcal{C}$ is the family of all congruence subgroups [Clo03].

######  Definition.

The set $\mathcal{R}$ of equivalence classes of unitary representations is
called _hereditary_ if

  1. (1)

$\mathcal{R}$ is _subrepresentation-closed_ : if $\rho\in\mathcal{R}$ and
$\pi\leq\rho$ then $\pi\in\mathcal{R}$,

  2. (2)

$\mathcal{R}$ is _dual-closed_ : if $\rho\in\mathcal{R}$ then the dual
representation satisfies $\rho^{*}\in\mathcal{R}$,

  3. (3)

$\mathcal{R}$ is _diagonal-closed_ : Let $\Pi$ be a representation of the
product group $\Gamma\times\Gamma$ whose restriction to each factor is
contained in $\mathcal{R}$. Then the restriction of $\Pi$ to the diagonal
embedding of $\Gamma$ is also contained in $\mathcal{R}$.

The set of finite-dimensional representations $\mathrm{FD}$ is hereditary. The
set of unitary representations $\mathcal{R}(\mathcal{N})$ associated to the
family $\mathcal{N}$ of normal subgroups is hereditary provided that
$\mathcal{N}$ is closed under taking finite intersections.

### Convergence of traces

Let $\mathcal{R}$ be a fixed _hereditary_ set of equivalence classes of
unitary representations of the group $\Gamma$. Denote

(5.2)
$\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)=\\{\varphi\in\mathrm{Ch}\left(\Gamma\right)\>:\>\pi_{\varphi}\in\mathcal{R}\\}.$

Note that the dual-closed and diagonal-closed conditions (which are part of
the notion of a hereditary set) imply that any character
$\varphi\in\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)$ has in addition
$\pi_{\varphi}\otimes\pi_{\varphi}^{*}\in\mathcal{R}$ as well as
$c_{\varphi}\in\mathcal{R}$.

Fix an arbitrary element $a\in\ell^{1}(\Gamma)$ satisfying $\|a\|_{1}=1$ and
such that the support of $a$ generates the group $\Gamma$. Note that the group
$\Gamma$ has property $\mathrm{T}(\mathcal{R})$ if and only if there is some
$0<\beta<1$ such that $\|\pi_{0}(a)\|\leq\beta$ for all $\pi\in\mathcal{R}$.

######  Proposition 5.1.

If the group $\Gamma$ has property $\mathrm{T}(\mathcal{R})$ then
$\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}\subset\mathrm{Ch}\left(\Gamma\right)$.

###### Proof.

Consider the family of the conjugation representations $c_{\varphi}$
corresponding to the characters
$\varphi\in\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)$. The subspace of
$c_{\varphi}$-invariant vectors for each such representation $c_{\varphi}$ is
one-dimensional and is spanned by the cyclic vector
$v_{\varphi}\in\mathcal{H}_{\varphi}$, see Corollary 3.2. The property
$\mathrm{T}(\mathcal{R})$ assumption implies that there is a fixed constant
$0<\beta<1$ (depending on the chosen element $a\in\ell^{1}(\Gamma)$) such that
operator norm bound
$\|c_{\varphi}(a)_{|\mathrm{span}_{\mathbb{C}}(v_{\varphi})^{\perp}}\|\leq\beta$
holds true for all characters
$\varphi\in\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)$. The same operator norm
bound holds true for any given limiting trace
$\psi\in\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$ by Proposition
4.2. In particular, the subspace of invariant vectors in the unitary
representation $c_{\psi}$ corresponding to the trace $\psi$ is spanned by the
cyclic vector $v_{\psi}$. We conclude that
$\psi\in\mathrm{Ch}\left(\Gamma\right)$ by the converse direction of Corollary
3.2. ∎

######  Proposition 5.2.

If the group $\Gamma$ has property $\mathrm{T}(\mathcal{R})$ then the trivial
character $1\in\mathrm{Ch}\left(\Gamma\right)$ is an isolated point of the set
$\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)\cup\\{1\\}$.

###### Proof.

Consider the family of the unitary representations $\pi_{\varphi}$
corresponding to the characters
$\varphi\in\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)\setminus\\{1\\}$. The
representations $\pi_{\varphi}$ admit no invariant vectors, see Corollary 2.2.
The property $\mathrm{T}(\mathcal{R})$ assumption implies that there is a
fixed constant $0<\beta<1$ (depending on the chosen element
$a\in\ell^{1}(\Gamma)$) such that the operator norm bound
$\|\pi_{\varphi}(a)\|\leq\beta$ holds true for all characters
$\varphi\in\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)\setminus\\{1\\}$. The
same operator norm bound holds true for any given limiting trace
$\psi\in\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)\setminus\\{1\\}}$
by Lemma 4.1. In particular, the unitary representation $\pi_{\psi}$
corresponding to the trace $\psi$ admits no invariant vectors. Moreover
$\psi\in\mathrm{Ch}\left(\Gamma\right)$ by Proposition 5.1. We conclude that
$\psi\neq 1$ by the converse direction of Corollary 2.2. ∎

Recall the notion of _amenable characters_ defined in the last paragraph of
§2.

######  Proposition 5.3.

Assume that the group $\Gamma$ has property $\mathrm{T}(\mathcal{R})$. Let
$\varphi_{n}\in\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)$ be any sequence of
characters converging to some $\varphi\in\mathrm{Tr}\left(\Gamma\right)$.
Assume that the characters $\varphi_{n}$ are either finite-dimensional and
satisfy $\dim_{\mathbb{C}}\mathcal{H}_{\varphi_{n}}\to\infty$ or are infinite-
dimensional. Then the character $\varphi$ is non-amenable.

###### Proof.

First, note that $\varphi\in\mathrm{Ch}\left(\Gamma\right)$ by Proposition
5.1. Second, the representation $\pi_{\varphi}$ must be infinite dimensional
by Proposition 5.7 below. The property $\mathrm{T}(\mathcal{R})$ assumption
implies that there is a fixed constant $0<\beta<1$ (depending on the chosen
element $a\in\ell^{1}(\Gamma)$) such that

  1. (1)

if the characters $\varphi_{n}$ are infinite-dimensional then the operator
norm bound $\|\pi_{\varphi_{n}}\otimes\pi_{\varphi_{n}}^{*}(a)\|\leq\beta$
holds true for each $n\in\mathbb{N}$, and

  2. (2)

if the characters $\varphi_{n}$ are finite-dimensional then the operator norm
bound
$\|\pi_{\varphi_{n}}\otimes\pi_{\varphi_{n}}^{*}(a)_{|\mathrm{span}_{\mathbb{C}}(\mathrm{Id}_{n})^{\perp}}\|\leq\beta$
holds true for each $n\in\mathbb{N}$.

We may apply Lemma 4.3 in the first case, or Lemma 4.3 in tandem with Lemma
4.4 in the second case, to deduce that the operator norm bound passes to the
limiting character $\varphi$, in the sense that
$\|(\pi_{\varphi}\otimes\pi_{\varphi}^{*})(a)\|\leq\beta<1$. Therefore the
limiting character $\varphi$ is non-amenable by definition. ∎

### Faces and Bauer simplices

Assume that the countable group $\Gamma$ has property
$\mathrm{T}(\mathcal{R})$. Then
$\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}\subset\mathrm{Ch}\left(\Gamma\right)$
by Proposition 5.1. To complete the proof of Theorem 1.5 of the introduction
we will require the following abstract observation.

######  Lemma 5.4.

Let $S$ be a metrizable Choquet simplex. If $X$ is a compact subset of the set
of extreme points $\partial S$ then $\overline{\mathrm{conv}\left(X\right)}$
is a face of the simplex $S$ and is a Bauer simplex.

###### Proof.

Denote $C=\overline{\mathrm{conv}\left(X\right)}$ so that $C$ is closed and
convex. Note that

(5.3) $X\subseteq C\cap\partial S\subseteq\partial C\subseteq\overline{X}.$

The rightmost inclusion in Equation (5.3) follows from Milman’s converse to
the Krein–Milman theorem [Mil47]. Since $X$ is compact $X=\overline{X}$ and
therefore equality must hold throughout. In particular $\partial C=X$.

We claim that the subset $C$ is a face of the simplex $S$. Suppose that $\mu$
is a probability measure with $\mathrm{supp}(\mu)\subset\partial S$ such that
its barycenter $\mathrm{bar}(\mu)=c$ satisfies $c\in C$. By Choquet’s theorem
applied to the convex set $C$ there exists a probability measure $\nu$ with
$\mathrm{supp}(\nu)\subset\partial C=X$ whose barycenter is
$\mathrm{bar}(\nu)=c$. Since $S$ is a Choquet simplex it follows that
$\mu=\nu$. In particular the measure $\mu$ is supported on $\partial C$ so
that $C$ is a face by definition.

To see that the convex set $C$ is a Choquet simplex note that as $\partial
C\subset\partial S$ any two probability measures supported on $\partial C$ are
in particular supported on $\partial S$. Therefore if these two measures have
the same barycenter in $C$ they must coincide. Finally the simplex $C$ is
Bauer since $\partial C=X$ is compact. ∎

###### Proof of Theorem 1.5.

We have already established that
$\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}\subset\mathrm{Ch}\left(\Gamma\right)$
in Proposition 5.1. The second part of the theorem saying that
$\overline{\mathrm{conv}\,\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$ is a
closed face of the simplex $\mathrm{Tr}\left(\Gamma\right)$ and is a Bauer
simplex follows abstractly by applying Lemma 5.4 with respect to the subset
$X=\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$ of the Choquet
simplex $S=\mathrm{Tr}\left(\Gamma\right)$. ∎

### Finite-dimensional representations

Recall that $\Gamma$ is a countable group and $\mathcal{R}$ is a hereditary
set of equivalence classes of unitary representations of the group $\Gamma$.
We establish a finiteness result for finite-dimensional characters in the
presence of property $\mathrm{T}(\mathcal{R})$, see Corollary 5.8.

######  Lemma 5.5.

Assume that the countable group $\Gamma$ has property
$\mathrm{T}(\mathcal{R})$. Let $\rho\in\overline{\mathcal{R}}$ be a finite-
dimensional unitary representation of the group $\Gamma$. Then

  1. (1)

$\rho\in\overline{\\{\pi\in\mathcal{R}\>:\>\text{$\pi$ has a non-trivial
finite-dimensional subrepresentation}\\}}$.

  2. (2)

If the representation $\rho$ is irreducible and
$\rho\in\overline{\mathcal{R}\cap\widehat{\Gamma}}$ then $\rho\in\mathcal{R}$
and $\rho$ is an isolated point of the subset
$\overline{\mathcal{R}\cap\widehat{\Gamma}}$ in the Fell topology.

###### Proof.

Let $\rho\in\overline{\mathcal{R}}$ be a finite-dimensional unitary
representation. Consider a sequence $\pi_{n}\in\mathcal{R}$ of unitary
representations such that $\pi_{n}\to\rho$ in the Fell topology. Therefore
$\rho\otimes\pi_{n}^{*}\to\rho\otimes\rho^{*}$ in the Fell topology. As the
unitary representation $\rho$ is finite-dimensional it satisfies
$1\leq\rho\otimes\rho^{*}$. In particular $\rho\otimes\pi_{n}^{*}\to 1$ in the
Fell topology. It follows that there is a subsequence $m_{n}$ such that
$\pi_{m_{n}}\otimes\pi^{*}_{n}\to 1$ in the Fell topology. Note that
$\pi_{m_{n}}\otimes\pi_{n}^{*}\in\mathcal{R}$ for all $n\in\mathbb{N}$ as the
family $\mathcal{R}$ is diagonal-closed. Property $\mathrm{T}(\mathcal{R})$
implies that $1\leq\pi_{m_{n}}\otimes\pi^{*}_{n}$ for all $n$ sufficiently
large. We conclude that the unitary representation $\pi_{n}$ admits a non-
trivial finite-dimensional sub-representation for all $n$ sufficiently large
[BdLHV08, Proposition A.1.12]. Item (1) follows.

Assume in addition that the unitary representations $\pi_{n}$ are all
irreducible. This additional assumption implies by Schur’s lemma that
$\pi_{n}$ is finite-dimensional and that $\pi_{n}=\pi_{m_{n}}$ for all
$n\in\mathbb{N}$. However, we may repeat the argument of the previous
paragraph with respect to the subsequence $m^{\prime}_{n}=m_{n}+1$. This
implies by Schur’s lemma that the sequence of unitary representations
$\pi_{n}$ stabilizes, namely there is some unitary representation
$\pi\in\mathcal{R}$ with $\pi_{n}=\pi$ for all $n\in\mathbb{N}$ sufficiently
large. The fact that the sequence $\pi_{n}$ converges to the representation
$\rho$ in the Fell topology is therefore equivalent to saying that
$\rho\prec\pi$. Since the representation $\pi$ is finite-dimensional and the
representation $\rho$ is irreducible, weak containment implies containment
[BdLHV08, Corollary F.2.9]. Therefore $\rho\leq\pi$. As the representation
$\pi$ is irreducible we conclude that $\rho=\pi\in\mathcal{R}$ and that $\rho$
is an isolated point of the subset
$\overline{\mathcal{R}\cap\widehat{\Gamma}}$ in the Fell topology. Item (2)
follows. ∎

Prior to proceeding we observe that the unitary representation given by the
GNS construction depends continuously on the trace in question.

######  Lemma 5.6.

Let $\mathrm{Rep}_{\aleph_{0}}(\Gamma)$ denote the set of equivalence classes
of separable unitary representations of the countable group $\Gamma$. The map

(5.4)
$\mathrm{Tr}\left(\Gamma\right)\to\mathrm{Rep}_{\aleph_{0}}(\Gamma),\quad\varphi\mapsto\pi_{\varphi}\quad\forall\varphi\in\mathrm{Tr}\left(\Gamma\right)$

is continuous with respect to the pointwise convergence topology on
$\mathrm{Tr}\left(\Gamma\right)$ and the Fell topology on
$\mathrm{Rep}_{\aleph_{0}}(\Gamma)$.

Essentially this lemma is a restatement in a different language of [BdLHV08,
Lemma F.1.3]. We explain this connection.

###### Proof of Lemma 5.6.

Consider a sequence of traces $\varphi_{n}\in\mathrm{Tr}\left(\Gamma\right)$
such that $\varphi_{n}\to\varphi$ pointwise for some trace
$\varphi\in\mathrm{Tr}\left(\Gamma\right)$. We wish to show that the sequence
of separable unitary representations $\pi_{\varphi_{n}}$ converges to the
unitary representation $\pi_{\varphi}$ in the Fell topology. This statement is
equivalent to saying that $\pi_{\varphi}\prec\bigoplus\pi_{\varphi_{n}}$.
Certainly the _particular_ positive definite function $\varphi$ associated to
the unitary representation $\pi_{\varphi}$ is a pointwise limit of the
positive definite functions $\varphi_{n}$ associated to the direct sum unitary
representation $\bigoplus\pi_{\varphi_{n}}$. However, in order to obtain the
desired weak containment it suffices by [BdLHV08, Lemma F.1.3] to consider
_only_ the particular positive definite function associated to the cyclic
vector $v_{\varphi}$. ∎

Certainly Lemma 5.6 applies equally well to the unitary representation
$\rho_{\varphi}$ in place of $\pi_{\varphi}$. The following is a restatement
of Proposition 1.7 from the introduction.

######  Proposition 5.7.

Assume that the countable group $\Gamma$ has property
$\mathrm{T}(\mathcal{R})$. Then any finite-dimensional character
$\varphi\in\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$ is an
isolated point of the subset
$\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$.

###### Proof.

Let $\varphi\in\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$ be a
finite-dimensional character. Consider a sequence of characters
$\varphi_{n}\in\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)$ satisfying
$\varphi_{n}\to\varphi$ in the pointwise convergence topology. The associated
unitary representations satisfy $\pi_{\varphi_{n}}\to\pi_{\varphi}$ in the
Fell topology according to Lemma 5.6. Moreover
$\pi_{\varphi_{n}}\in\mathcal{R}$ by assumption so that
$\pi_{\varphi}\in\overline{\mathcal{R}}$. Item (1) of Lemma 5.5 implies that
the representations $\pi_{\varphi_{n}}$ admit finite-dimensional sub-
representations. According to Lemma 2.3 and as the $\varphi_{n}$’s are
characters we observe that the $\varphi_{n}$’s are finite-dimensional for all
$n$ sufficiently large. So $\pi_{\varphi_{n}}=d_{n}\rho_{n}$ and
$\pi_{\varphi}=d\rho$ for some finite-dimensional irreducible unitary
representations $\rho_{n},\rho\in\widehat{\Gamma}$ where

(5.5)
$d_{n}=\dim\rho_{n},d=\dim\rho\quad\text{and}\quad\varphi_{n}=\frac{1}{d_{n}}\mathrm{tr}\rho_{n},\varphi=\frac{1}{d}\mathrm{tr}\rho.$

In this situation $\rho_{n}\to\rho$ in the Fell topology. It follows from Item
(2) of Lemma 5.5 that $\rho_{n}=\rho$ for all $n$ sufficiently large. So
$\varphi_{n}=\varphi$ for all $n$ sufficiently large. We conclude that the
character $\varphi$ is an isolated point as required. ∎

######  Corollary 5.8.

If the group $\Gamma$ has property $\mathrm{T}(\mathcal{R})$ then
$\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$ admits only finitely
many characters in each finite dimension.

###### Proof.

The subset $\widehat{\Gamma}_{\leq d}$ of the unitary dual consisting of the
equivalence classes of irreducible representations of dimension at most $d$ is
closed in the Fell topology for all $d\in\mathbb{N}$ [Fel60, Lemma 2.4]. The
Kolmogorov quotient999The _Kolmogorov quotient_ of a topological space is its
quotient by the equivalence relation of topological indistinguishability. In
other words the Kolmogorov quotient is the largest $T_{0}$ quotient. of the
unitary dual $\widehat{\Gamma}$ is the space $\mathrm{Prim}(\Gamma)$
consisting of weak equivalence classes of unitary representations [BdlH20,
1.E.3]. The quotient map to the Kolmogorov quotient is closed [Pir21,
Corollary 3.18]. Therefore the image of the subset $\widehat{\Gamma}_{\leq d}$
in $\mathrm{Prim}(\Gamma)$ is closed for all $d\in\mathbb{N}$. Pulling from
$\mathrm{Prim}(\Gamma)$ back to $\mathrm{Ch}\left(\Gamma\right)$ (via the
natural map described in [BdlH20, 7.F.2]) and relying on Lemma 5.6, we
conclude that the subspace of
$\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$ consisting of all
characters up to a given finite dimension is closed and hence compact. Each
character in $\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}$ is an
isolated point by Proposition 5.7. The finiteness of this subspace follows. ∎

######  Remark 5.9.

See [BLMM02, Proposition 2] as well as [LL08, Proposition 4.1] for a related
fact: finite abelianization implies finitely many representations factoring
through a finite index kernel in each dimension.

##  6\. Character limits of groups with property $\mathrm{T}(\mathcal{R})$

We study groups enjoying various flavours of property
$\mathrm{T}(\mathcal{R})$ and their character theory. The information gathered
in the previous section §5 leads to the following general statement on
vanishing of characters.

######  Theorem 6.1.

Let $\Gamma$ be a countable group and $N\lhd\Gamma$ a normal subgroup. Let
$\mathcal{R}$ be a hereditary set of equivalence classes of unitary
representations of $\Gamma$. Assume

  1. (1)

_(spectral gap)_ the group $\Gamma$ has property $\mathrm{T}(\mathcal{R})$,
and

  2. (2)

_(character dichotomy)_ any character of the group $\Gamma$ is either amenable
or vanishes outside the subgroup $N$.

Let $\varphi_{n}\in\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)$ by any sequence
of pairwise distinct characters. Then any accumulation point of the sequence
$\varphi_{n}$ is a character supported on the subgroup $N$. Namely

(6.1)
$\quad\lim_{n\to\infty}\varphi_{n}(\gamma)=0\quad\forall\gamma\in\Gamma\setminus
N.$

###### Proof.

Recall that the space $\mathrm{Tr}\left(\Gamma\right)$ is compact. Let
$\varphi\in\mathrm{Tr}\left(\Gamma\right)$ be any accumulation point of the
sequence $\varphi_{n}$. We know from Proposition 5.1 that
$\overline{\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)}\subset\mathrm{Ch}\left(\Gamma\right)$.
In particular the accumulation point $\varphi$ is a _character_ of the group
$\Gamma$.

The subset $\mathrm{Ch}\left(\Gamma;\mathcal{R}\right)$ contains only finitely
many distinct characters in each dimension, see Corollary 5.8. We may assume
up to passing to a subsequence either that the characters $\varphi_{n}$ are
all infinite-dimensional or are all finite-dimensional and the dimension
$\dim_{\mathbb{C}}\mathcal{H}_{\varphi_{n}}$ tends to infinity. In either case
we conclude from Proposition 5.3 that the accumulation point $\varphi$ is a
_non-amenable_ character. The character dichotomy assumption implies that
$\varphi(\gamma)=0$ for all elements $\gamma\in\Gamma\setminus N$. Since
$\varphi$ was taken to be an arbitrary accumulation point of the sequence
$\varphi_{n}$ in the topology of pointwise convergence Equation (6.1) holds
true. ∎

The general principle formulated in Theorem 6.1 can be specialized to obtain
the following useful result.

######  Theorem 6.2.

Let $\Gamma$ be a group with property $\mathrm{T}(\mathrm{FD})$. Assume that
every character of $\Gamma$ is either finite-dimensional or vanishes outside
the center $\mathrm{Z}(\Gamma)$. If
$\varphi_{n}\in\mathrm{Ch}\left(\Gamma\right)$ is any sequence of pairwise
distinct characters then $\varphi_{n}(\gamma)\rightarrow 0$ for all elements
$\gamma\in\Gamma\setminus\mathrm{Z}(\Gamma)$.

###### Proof.

The asymptotic vanishing result in question follows immediately by applying
Theorem 6.1 with respect to the family $\mathcal{R}=\mathrm{FD}$ and where the
normal subgroup $N$ is taken to be the center $\mathrm{Z}(\Gamma)$ of the
group $\Gamma$. ∎

Let us explain how Theorem 6.2 implies our main result, namely Theorem 1.1 of
the introduction.

###### Proof of Theorem 1.1.

Let $G$ be a semisimple Lie group with finite center. Assume that
$\mathrm{rank}_{\mathbb{R}}(G)\geq 2$ and that some non-compact simple factor
of $G$ has property (T). Let $\Gamma$ be an irreducible lattice in the Lie
group $G$. Every character of $\Gamma$ is either finite-dimensional of
vanishes outside the center of $\Gamma$ by [BBH21]. Furthermore, the lattice
$\Gamma$ has property T(FD) by [LZ89]. This means that Theorem 6.2 applies. ∎

In the next theorem we consider the situation in Example (2) on p. 2.

######  Theorem 6.3.

Let $k$ be an algebraic number field with ring of integers $\mathcal{O}_{k}$
and $S\subset\mathcal{O}_{k}$ be a finite (possibly empty) multiplicative
subset such that the localization $S^{-1}\mathcal{O}_{k}$ has infinitely many
units. Consider the linear group
$\Gamma=\mathrm{SL}_{2}(S^{-1}\mathcal{O}_{k})$. If
$\varphi_{n}\in\mathrm{Ch}\left(\Gamma\right)$ is any sequence of pairwise
distinct characters then $\varphi_{n}(\gamma)\to 0$ for all elements
$\gamma\in\Gamma\setminus\mathrm{Z}(\Gamma)$

###### Proof.

Every character of the linear group $\Gamma$ is either finite-dimensional or
vanishes outside the center [PT16, Theorem 2.6]. In fact, all finite-
dimensional characters of the group $\Gamma$ come from finite quotients, and
all finite quotients are congruence quotients [Ser70]. Selberg’s property
[Clo03] implies that the group $\mathrm{SL}_{2}(S^{-1}\mathcal{O}_{k})$ has
property T(FD). As such Theorem 6.2 applies. ∎

Next, we deal with the case of congruence characters, i.e. characters
factoring through a congruence subgroup. The following justifies Example (3)
on p. 3.

######  Theorem 6.4.

Let $G$ be a semisimple Lie group with $\mathrm{rank}_{\mathbb{R}}(G)\geq 2$
and finite center. Let $\Gamma$ be an irreducible lattice in $G$ and
$\varphi_{n}\in\mathrm{Ch}\left(\Gamma\right)$ be any sequence of pairwise
distinct _congruence_ characters. Then $\varphi_{n}(\gamma)\to 0$ for all
elements $\gamma\in\Gamma\setminus\mathrm{Z}(\Gamma)$.

###### Proof.

Let $\Gamma$ be an irreducible lattice in a higher rank semisimple Lie group.
The lattice $\Gamma$ is arithmetic by the Margulis arithmeticity theorem. The
arithmetic group $\Gamma$ has Selberg’s property [Clo03]. In our terminology
this is the same as property $\mathrm{T}(\mathcal{R}(\mathcal{C}))$ where
$\mathcal{C}$ is the family of congruence subgroups of the lattice $\Gamma$.
Any non-amenable character of the group $\Gamma$ vanishes outside of its
center $\mathrm{Z}(\Gamma)$ by [BBHP22, Theorem A]. The desired vanishing
result follows as a special case of the general Theorem 6.1 with respect to
the family $\mathcal{R}=\mathcal{R}(\mathcal{C})$ and where the normal
subgroup $N$ is taken to be the center $\mathrm{Z}(\Gamma)$ of the group
$\Gamma$. ∎

### Generalization to local fields

Theorem 1.1 can be generalized to $S$-arithmetic groups over local fields. We
now provide some additional details.

Let $K$ be a global field with ring of integers $\mathcal{O}$. Denote by
$V^{K}$ denote the set of all equivalence classes of valuations on the field
$K$ up to equivalence and by $V^{K}_{\infty}\subset V^{K}$ the subset of
Archimedean valuations. Let $S\subset V^{K}$ be a finite set of valuations
with $V^{K}_{\infty}\subset S$. The corresponding localization of the ring of
integers $\mathcal{O}$ is

(6.2) $\mathcal{O}_{S}=\\{x\in K\>:\>|x|_{s}\leq 1\quad\forall s\in
V^{K}\setminus S\\}.$

Let $\mathbf{G}$ be a connected almost $K$-simple algebraic $K$-group. Fix an
injective $K$-representation $\rho:\mathbf{G}\to\mathrm{GL}_{N}$ for some
$N\in\mathbb{N}$. Denote

(6.3) $\Gamma_{S}=\rho^{-1}(\mathrm{GL}_{N}(\mathcal{O}_{S})).$

A subgroup $\Gamma$ of $\mathbf{G}(K)$ is called _$S$ -arithmetic_ if it is
commensurable with $\Gamma_{S}$. This notion is independent of the choice of
the representation $\rho$. The $S$-arithmetic subgroup $\Gamma$ is said to be
of

  * •

_compact type_ if $\rho(\Gamma_{S})$ is compact in $\mathbf{G}(K_{v})$ for
every absolute value $v$ on $K$,

  * •

_simple type_ if there exists a unique absolute value $v$ on $K$ such that
$\rho(\Gamma_{S})$ is unbounded, and

  * •

_product type_ otherwise.

The $S$-arithmetic subgroup $\Gamma$ is called _higher rank_ if $\Gamma$ has
simple type and $\mathrm{rank}_{K_{v}}(\mathbf{G})\geq 2$ or if $\Gamma$ has
product type.

######  Theorem 6.5.

Let $\Gamma$ be a higher rank $S$-arithmetic subgroup. Assume that the group
$\mathbf{G}(K_{v})$ is non-compact and has property $(T)$ for some absolute
value $v$ on $K$. Let $\varphi_{n}\in\mathrm{Ch}\left(\Gamma\right)$ be any
sequence of pairwise distinct characters. Then $\varphi_{n}(\gamma)\to 0$ for
all $\gamma\in\Gamma\setminus\mathrm{Z}(\Gamma)$.

In particular, if the $S$-arithmetic group $\Gamma$ is center-free then the
conclusion of Theorem 6.5 says that $\varphi_{n}\to\delta_{e}$ in the
pointwise convergence topology.

###### Proof of Theorem 6.5.

If $\Gamma$ has simple type then it has Kazhdan’s property (T) [BdLHV08,
Theorem 1.6.1]. If $\Gamma$ has product type then the assumption that
$\mathbf{G}(K_{v})$ has property (T) for some absolute value $v$ on the global
field $K$ guarantees that $\Gamma$ has property $\mathrm{T}(\mathrm{FD})$ by
[LZ89, Corollary 2.6.(i)]. To be able to apply [LZ89] we also need to use the
fact that any non-compact group of the form $\mathbf{G}(K_{v})$ is _minimally
almost periodic_ , i.e. admits no non-trivial finite dimensional unitary
representations [BG17]. Higher rank $S$-arithmetic groups have the character-
dichotomy property by [BBH21, Theorem B]. At this point the desired conclusion
follows from our general Theorem 6.1. ∎

This theorem has been mentioned in Example (1) on p. 1 following Theorem 1.1.

### Finite groups of Lie type

Let $\mathfrak{g}$ be a complex simple101010This entire subsection makes sense
for semisimple rather than simple Lie algebras. We restrict attention to the
simple case since it suffices for our purposes. Lie algebra. The Lie algebra
$\mathfrak{g}$ admits a _Chevalley basis_ $B$. The basis $B$ is unique up to
automorphisms of the Lie algebra $\mathfrak{g}$ and sign changes [Ste67]. Let
$\mathfrak{g}_{\mathbb{Z}}$ be the $\mathbb{Z}$-span of the Chevalley basis
$B$ so that $\mathfrak{g}_{\mathbb{Z}}$ is a lattice in the complex vector
space $\mathfrak{g}$.

Let $K$ be a field. The _Chevalley group_ $G(K)$ associated to the Lie algebra
$\mathfrak{g}$ is a connected $K$-simple adjoint111111The general construction
of a Chevalley group depends on a choice of a lattice containing the root
lattice and contained in the lattice of weights. For our purposes it will
suffice to consider the root lattice, which gives rise to adjoint Chevalley
groups. $K$-subgroup of the group
$\mathrm{GL}(\mathfrak{g}_{\mathbb{Z}}\otimes K)\cong\mathrm{GL}_{N}(K)$ where
$N=\dim_{\mathbb{C}}(\mathfrak{g})$. The algebraic $K$-group $G(K)$ is
$K$-split and its $K$-rank is equal to the rank of the complex simple Lie
algebra $\mathfrak{g}$.

Assume from now on that $K$ is a global field. Consider the localization
$\mathcal{O}_{S}$ of its ring of integers $\mathcal{O}$ corresponding to some
finite subset $S$ of non-Archimedean valuations as in Equation 6.2. Regard the
Chevalley group $G(K)$ equipped with the rational representation defined via
the Chevalley basis $B$. In particular the subgroup $G(\mathcal{O}_{S})$
consisting of the $\mathcal{O}_{S}$-rational points of the group $G(K)$ is the
stabilizer of the submodule $\mathfrak{g}_{\mathbb{Z}}\otimes\mathcal{O}_{S}$.
The group $G(\mathcal{O}_{S})$ is $S$-arithmetic by definition.

Consider any non-zero ideal $\mathcal{I}\leq\mathcal{O}_{S}$ such that
$F=\mathcal{O}_{S}/\mathcal{I}$ is a finite ring (or field). Let $G(F)$ be the
Chevalley group over the ring $F$. Since the ring $F$ is finite the Chevalley
group $G(F)$ is known to coincide with its elementary subgroup generated by
the root subgroups [Wei13, Lemma III.1.4]. This fact implies that the natural
homomorphism $f_{\mathcal{I}}:G(\mathcal{O}_{S})\to G(F)$ is surjective
[Wei13, Remark III.1.2.3].

We utilize the discussion in the preceding paragraphs to obtain character
limit results for finite simple groups of Lie type.

###### Proof of Corollary 1.2.

Let $G$ be a Chevalley group associated to some complex simple Lie algebra of
rank two or higher. There are two families of fields to consider.

In the case where the global field $K$ is a finite extension of $\mathbb{Q}$
we consider the local field $k=K_{\infty}$ which is either $\mathbb{R}$ or
$\mathbb{C}$.

In the case where $K$ is a finite extension of the field of rational function
over a finite field $\mathbb{F}_{q}$, we may identify (up to an isomorphism)
the ring $\mathcal{O}_{K}$ with the localization
$\mathcal{O}_{S}=\mathbb{F}_{q}\left[t^{-1}\right]$ of the ring of algebraic
integers $\mathcal{O}_{K}$ associated to the subset of valuations $S=\\{t\\}$.
We consider the local field
$k=K_{t}=\mathbb{F}_{p}\left(\left(t\right)\right)$.

In either case the $k$-analytic group $G(k)$ has $k$-rank at least two. In
particular the group $G(k)$ is non-compact and has property $(T)$. In
particular, Theorem 6.5 applies to the lattice $G(\mathcal{O}_{K})$.

To conclude the proof, consider an arbitrary enumeration $\varphi_{n}$ of all
characters of the group $G(\mathcal{O}_{K})$ which factorize through some
finite quotient of the form $G(\mathcal{O}_{K}/I)$ where $I$ is an ideal in
$\mathcal{O}_{K}$. According to Theorem 6.5, for any $\varepsilon>0$ there is
some sufficiently large $N>0$ such that $|\varphi_{n}(\gamma)|<\varepsilon$
for any non-central element $\gamma\in G(\mathcal{O}_{K})$. The desired
statement follows by excluding finitely many ideals $I\lhd\mathcal{O}_{K}$. ∎

### Generalization to non-semisimple groups

Any discrete group $\Gamma$ admits a maximal amenable normal subgroup
$\mathrm{Rad}(\Gamma)$ called the _amenable radical_. The following statement
generalizes Theorem 1.1 to the non-semisimple setting; see Example (4) on page
4.

######  Theorem 6.6.

Let $\mathbf{G}$ be a connected $\mathbb{Q}$-algebraic group with solvable
radical $\mathbf{R}$. Assume that the semisimple group $\mathbf{G}/\mathbf{R}$
has at most one $\mathbb{Q}$-simple $\mathbb{R}$-anistorpic factor and no
$\mathbb{R}$-simple factors of rank one. Let
$\Gamma\leq\mathbf{G}(\mathbb{Q})$ be an arithmetic subgroup with property
(T). Let $\varphi_{n}\in\mathrm{Ch}\left(\Gamma\right)$ be any sequence of
pairwise distinct characters. Then $\varphi_{n}(g)\to 0$ for any element
$\gamma\in\Gamma\backslash\mathrm{Rad}(\Gamma)$.

###### Proof.

The crucial ingredient here is [BV22, Theorem A]. Taking into account the
property (T) assumption, it says that any character of the arithmetic group
$\Gamma$ is either finite-dimensional or vanishes outside the amenable radical
$\mathrm{Rad}\left(\Gamma\right)$. We conclude by relying on our general
Theorem 6.1, taking $\mathcal{R}$ to be the set of equivalence classes of all
separable unitary representations and letting
$N=\mathrm{Rad}\left(\Gamma\right)$. ∎

### Convergence of representations

We prove the results stated in the introduction concerning convergence of
representations in the Fell topology.

######  Definition.

Let $\pi_{1}$ and $\pi_{2}$ be two unitary representations of a group
$\Gamma$.

  * •

$\pi_{1}$ is _subordinate_ to $\pi_{2}$ if every non-trivial subrepresentation
of $\pi_{1}$ contains a non-trivial subrepresentation equivalent to a
subrepresentation of $\pi_{2}$.

  * •

$\pi_{1}$ and $\pi_{2}$ are _quasi-equivalent_ if each one is subordinate to
the other.

  * •

Quasi-equivalence is denoted $\pi_{1}\approx\pi_{2}$.

Note that a finite-dimensional unitary representation $\pi_{1}$ is subordinate
to a unitary representation $\pi_{2}$ if every irreducible subrepresentation
of $\pi_{1}$ is equivalent to a subrepresentation of $\pi_{2}$.

###### Proof of Theorem 1.3.

Let $\Gamma$ be a center-free irreducible lattice as in Theorem 1.3. As was
already mentioned above, the lattice $\Gamma$ has property
$\mathrm{T}(\mathrm{FD})$ by [LZ89] and every character of $\Gamma$ other than
$\delta_{\\{e\\}}$ is finite-dimensional by [BBH21].

Consider a sequence $\pi_{n}$ of finite-dimensional unitary representations of
the group $\Gamma$. Assume that the representations $\pi_{n}$ are pairwise non
quasi-equivalent. Note that for every finite-dimensional unitary
representation $\pi$, there are only finitely many quasi-equivalence classes
of finite-dimensional unitary representations subordinate to $\pi$. Therefore,
up to passing to a subsequence, we may find for each $n\in\mathbb{N}$ an
_irreducible_ subrepresentation $\sigma_{n}\leq\pi_{n}$ such that the
representations $\sigma_{n}$ are pairwise non-equivalent. It follows that the
characters $\varphi_{n}=\frac{1}{\dim\sigma_{n}}\mathrm{tr}\circ\sigma_{n}$
are pairwise distinct. At this point we invoke Theorem 6.2. It follows that
the sequence of characters $\varphi_{n}$ converges to the Dirac trace
$\delta_{e}$ pointwise. Note that the GNS construction associated to the trace
$\delta_{e}$ is the regular representation $\lambda_{\Gamma}$. Recall that the
GNS construction depends continuously on the trace, see Lemma 5.6. Hence the
sequence of unitary representations $\sigma_{n}$, and a fortiori also the
sequence of representations $\pi_{n}$, converges to the regular representation
$\lambda_{\Gamma}$ in the Fell topology. ∎

###### Proof of Corollary 1.4.

Let $\Gamma$ be a center-free lattice as in Theorem 1.3. We will prove that
the sequence of quasi-regular representations $L^{2}(\Gamma/\Gamma_{n})$
converges to the regular representation $\lambda_{\Gamma}$ in the Fell
topology by showing that every subsequence contains a further subsequence with
limit $\lambda_{\Gamma}$. Let $n_{k}$ be an arbitrary subsequence. We claim
that the sequence of unitary representations $L^{2}(\Gamma/\Gamma_{n_{k}})$
contains a further subsequence consisting of pairwise non quasi-equivalent
representations. If this is not the case then
$\bigoplus_{k}L^{2}(\Gamma/\Gamma_{n_{k}})$ is quasi-equivalent to some
finite-dimensional representation $\rho$ with $\ker\rho=N$ satisfying
$\left[\Gamma:N\right]<\infty$. Quasi-equivalent representations have the same
kernel [BdlH20, Proposition 6.A.4]. In particular $N\leq\Gamma_{n_{k}}$ for
all $k\in\mathbb{N}$. This leads to a contradiction to the fact the sequence
$\Gamma_{n_{k}}$ is strictly decreasing. Having established the claim, the
convergence of this further subsequence to the representation
$\lambda_{\Gamma}$ follows from Theorem 1.3. ∎

We also identify all possible limits of sequences of finite-dimensional
unitary representations of lattices.

######  Proposition 6.7.

Let $\Gamma$ be a center-free irreducible lattice as in Theorem 1.3. Let
$\pi_{n}$ be a sequence of pairwise disjoint finite-dimensional unitary
representations of $\Gamma$. Then any limit of the sequence $\pi_{n}$ in the
Fell topology weakly contains the regular representation $\lambda_{\Gamma}$.

###### Proof.

Let $\pi$ be any limit point of the sequence of representations $\pi_{n}$ in
the Fell topology. We claim that the representation $\pi$ is _weakly mixing_ ,
i.e. does not contain any finite-dimensional subrepresentations. Assume by
contradiction that $\pi$ does contain some irreducible finite-dimensional
subrepresentation $\rho$. Up to passing to a subsequence, we may assume that
$\rho\not\leq\pi_{n}$ for all $n\in\mathbb{N}$. Let $\sigma_{k}$ be some
enumeration of all finite-dimensional irreducible subrepresentations contained
in some representation $\pi_{n}$. Then

(6.4) $\rho<\pi\prec\bigoplus_{n}\pi_{n}\approx\bigoplus_{k}\sigma_{k}$

In particular, the irreducible finite-dimensional representations $\sigma_{k}$
converge to the representation $\rho$ in the Fell topology. The lattice
$\Gamma$ has property T(FD) by [LZ89]. This is a contradiction to Lemma 5.5.
As the representation $\pi$ is weakly mixing, it follows from [BBH21, Theorem
B] and from [BBHP22, Proposition 3.6] that $\pi$ weakly contains the regular
representation $\lambda_{\Gamma}$. ∎

We remark that the above theorems concerning the convergence of
representations naturally extend to semisimple algebraic groups defined over
arbitrary local fields and their $S$-arithmetic subgroups; see the discussion
in Theorem 6.5.

##  7\. Relative traces and characters

Let $\Gamma$ be a countable discrete group. Let $\Lambda$ be another countable
discrete group admitting an action by automorphisms
$\theta:\Lambda\to\mathrm{Aut}\left(\Gamma\right)$ on the group $\Gamma$. This
induces an action of the group $\Lambda$ on the set of traces
$\mathrm{Tr}\left(\Gamma\right)$ by precomposition. The _relative traces_ of
the group $\Gamma$ are given by

(7.1)
$\mathrm{Tr}_{\Lambda}\left(\Gamma\right)=\\{\varphi\in\mathrm{Tr}\left(\Gamma\right)\>:\>\varphi^{\lambda}=\varphi\quad\forall\lambda\in\Lambda\\}.$

The set of relative traces $\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$ is a
simplex [BV22, Proposition 2.2]. Its extreme points are called _relative
characters_ and are denoted by $\mathrm{Ch}_{\Lambda}\left(\Gamma\right)$.

The set $\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$ stands in a bijective
correspondence with the set of all $\Lambda$-invariant Borel probability
measures on $\mathrm{Ch}\left(\Gamma\right)$ via the barycenter map. The
relative characters $\mathrm{Ch}_{\Lambda}\left(\Gamma\right)$ correspond to
ergodic $\Lambda$-invariant measures on the space
$\mathrm{Ch}\left(\Gamma\right)$.

Consider the situation where the group $\Lambda$ has Kazhdan’s property (T)
and $\Gamma$ is a normal subgroup of $\Lambda$. The group $\Lambda$ acts on
its subgroup $\Gamma$ by conjugation. If $\mathrm{Tr}\left(\Gamma\right)$
happens to be a Bauer simplex so that the space
$\mathrm{Ch}\left(\Gamma\right)$ is compact then it is an immediate
consequence of the above discussion, together with the theorem of Glasner and
Weiss [GW97], that the space of relative traces
$\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$ is also a Bauer simplex. For
example, this will be the case if the group $\Gamma$ is abelian or has
Kazhdan’s property (T) in its own right.

The main point of the remainder of this section will be to obtain a similar
conclusion assuming that the group $\Lambda$ has property (T) but without
assuming that the simplex $\mathrm{Tr}\left(\Gamma\right)$ is Bauer.

### Relative traces and the GNS data

Consider a relative trace $\varphi\in\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$
with corresponding GNS data
($\pi_{\varphi},\rho_{\varphi},\mathcal{H}_{\varphi},v_{\varphi})$. There is a
unique unitary representation
$U_{\varphi}:\Lambda\to\mathrm{U}(\mathcal{H}_{\varphi})$ given by
$U_{\varphi}:\lambda\mapsto U_{\varphi}(\lambda)$ which satisfies

(7.2)
$U_{\varphi}(\lambda)\pi_{\varphi}(\gamma)U_{\varphi}(\lambda)^{-1}=\pi_{\varphi}(\gamma^{\lambda})\quad\text{and}\quad
U_{\varphi}(\lambda)v_{\varphi}=v_{\varphi}\quad\forall\lambda\in\Lambda,\gamma\in\Gamma.$

Note that $U_{\varphi}(\lambda)=\pi_{\varphi}(\gamma)$ provided that
$\theta(\lambda)=\mathrm{Inn}(\gamma)$, namely the element $\lambda\in\Lambda$
acts on the group $\Gamma$ via the inner automorphism corresponding to the
element $\gamma\in\Gamma$. For all this see [BF20, Proposition 2.7].

Let $N_{\varphi}$ denote the von Neumann algebra generated by the operators
$\pi_{\varphi}(\Gamma)$ as well as the $U_{\varphi}(\lambda)$’s for all
elements $\lambda\in\Lambda$, namely

(7.3)
$N_{\varphi}=\left(\pi_{\varphi}(\Gamma)\;\cup\;U_{\varphi}(\Lambda)\right)^{\prime\prime}.$

Then $N_{\varphi}^{\prime}=\mathbb{C}\cdot\mathrm{Id}$ if and only if
$\varphi\in\mathrm{Ch}_{\Lambda}\left(\Gamma\right)$ [BF20, Proposition 2.8].

######  Proposition 7.1.

Assume that $\mathrm{Inn}(\Gamma)\leq\theta(\Lambda)$. Then any relative trace
$\varphi\in\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$ satisfies

(7.4) $N^{\prime}_{\varphi}=\mathrm{Z}(M_{\varphi})\cap
U_{\varphi}(\Lambda)^{\prime}.$

###### Proof.

The inclusion in the following direction is immediate

(7.5) $\mathrm{Z}(M_{\varphi})\cap
U_{\varphi}(\Lambda)^{\prime}\leq\pi_{\varphi}(\Gamma)^{\prime}\cap
U_{\varphi}(\Lambda)^{\prime}=N^{\prime}_{\varphi}.$

Let us argue in the converse direction and show that
$N^{\prime}_{\varphi}\leq\mathrm{Z}(M_{\varphi})\cap
U_{\varphi}(\Lambda)^{\prime}$. Consider any element $T\in
N^{\prime}_{\varphi}$. Then $\varphi_{T^{\frac{1}{2}}}$ is a
$\Lambda$-invariant positive-definite function on the group $\Gamma$ dominated
by the trace $\varphi$ [BF20, Proposition 2.8]. Since $\theta(\Lambda)$
contains all inner automorphisms,
$\varphi_{T^{\frac{1}{2}}}\in\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$. The
element $T$ with these properties is uniquely determined in
$N^{\prime}_{\varphi}$. On the other hand
$\varphi_{T^{\frac{1}{2}}}=\varphi_{S^{\frac{1}{2}}}$ for some element
$S\in\mathrm{Z}(M_{\varphi})$ with $0\leq S\leq 1$. The fact that
$\varphi_{T^{\frac{1}{2}}}=\varphi_{S^{\frac{1}{2}}}$ is $\Lambda$-invariant
implies that $S\in U_{\varphi}(\Lambda)^{\prime}$, see the first paragraph on
[BF20, p.11]. We obtain $S\in N^{\prime}_{\varphi}$. The uniqueness of the
element $T$ as above means that $T=S$ so that $T\in\mathrm{Z}(M_{\varphi})$.
As $T\in U_{\varphi}(\Lambda)^{\prime}$ the desired result follows. ∎

As was explained in §3, the mapping

(7.6)
$\iota_{\varphi}:M_{\varphi}\to\mathcal{H}_{\varphi},\quad\iota_{\varphi}:x\mapsto
xv_{\varphi}\quad\forall x\in M_{\varphi}$

allows us to regard the von Neumann algebra $M_{\varphi}$ as a dense subspace
of the Hilbert space $\mathcal{H}_{\varphi}$. For every element
$\lambda\in\Lambda$ consider the map

(7.7) $\alpha_{\varphi}(\lambda):M_{\varphi}\to
M_{\varphi},\quad\alpha_{\varphi}(\lambda)x=U_{\varphi}(\lambda)xU_{\varphi}(\lambda)^{-1}.$

For each element $\lambda\in\Lambda$ the operator $\alpha_{\varphi}(\lambda)$
preserves the norm of vectors in the image of the map $\iota_{\varphi}$, in
the sense that

(7.8)
$\|\iota_{\varphi}(x)\|_{\mathcal{H}_{\varphi}}=\|\iota_{\varphi}(\alpha_{\varphi}(\lambda)x)\|_{\mathcal{H}_{\varphi}}\quad\forall
x\in M_{\varphi}.$

As such the operator $\alpha_{\varphi}(\lambda)$ extends to a unitary operator
on the Hilbert space $\mathcal{H}_{\varphi}$. This process determines a
unitary representation
$\alpha_{\varphi}:\Lambda\to\mathrm{U}(\mathcal{H}_{\varphi})$. Observe that
an element $x\in M_{\varphi}$ satisfies $x\in U_{\varphi}(\Lambda)^{\prime}$
if and only if $\iota_{\varphi}(x)=xv_{\varphi}\in\mathcal{H}_{\varphi}$ is an
invariant vector for the unitary representation $\alpha_{\varphi}$.

######  Proposition 7.2.

Assume that $\mathrm{Inn}(\Gamma)\leq\Lambda$. Let
$\varphi\in\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$ be a relative trace. Then
$\varphi\in\mathrm{Ch}_{\Lambda}\left(\Gamma\right)$ if and only if
$\dim_{\mathbb{C}}\mathcal{H}_{\varphi}^{\alpha_{\varphi}}=1$.

###### Proof.

The same proof as that of Proposition 3.1 relying on Lemma 3.3 shows that the
commutant $N_{\varphi}^{\prime}$ is dense in the Hilbert subspace
$\mathcal{H}_{\varphi}^{\alpha_{\varphi}}$. Note that any element
$\lambda\in\Lambda$ which acts on the group $\Gamma$ via an inner automorphism
of the form $\mathrm{Inn}(\gamma)$ for some element $\gamma\in\Gamma$
satisfies $\alpha_{\varphi}(\lambda)=c_{\varphi}(\gamma)$ where
$c_{\varphi}:\Gamma\to\mathrm{U}(\mathcal{H}_{\varphi})$ is the conjugation
representation studied in §3. The desired conclusion follows by using the
aforementioned fact that the relative trace $\varphi$ is a relative character
if and only if $N^{\prime}_{\varphi}=\mathbb{C}\cdot\mathrm{Id}$. ∎

Assume that $\lambda\in\Lambda$ is an element satisfying
$\theta(\lambda)=\mathrm{Inn}(g)$ for some element $\gamma\in\Gamma$. In this
case we have already seen that $U_{\varphi}(\lambda)=\pi_{\varphi}(\gamma)$.
As a result $\alpha_{\varphi}(\lambda)=c_{\varphi}(\gamma)$ holds true.
Therefore Proposition 7.2 can be seen as a generalization to the relative
setting of the statements in §3 dealing with the conjugation representation.

### Spectral gap for the representation $\alpha_{\varphi}$

The action by automorphisms of the group $\Lambda$ on the group $\Gamma$
induces an action on its group ring $\mathbb{C}[\Gamma]$. We denote this
action by $x\mapsto x^{\lambda}$ for any element $\lambda\in\Lambda$ and
$x\in\mathbb{C}\left[\Gamma\right]$.

Fix an arbitrary element $a\in\ell^{1}(\Lambda)$ with $\|a\|_{1}=1$ whose
support generates the group $\Lambda$. Denote $b=a^{*}a$ and write
$b=\sum_{\lambda\in\Lambda}b_{\lambda}\lambda$ with coefficients
$b_{\lambda}\in\mathbb{C}$. We extend Lemma 4.2 to the relative setting.

######  Lemma 7.3.

Fix $0<\beta<1$. Let $\varphi\in\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$ be a
relative trace. Then

(7.9)
$\|\alpha_{\varphi}(a)_{|\mathrm{span}_{\mathbb{C}}(v_{\varphi})^{\perp}}\|\leq\beta^{\frac{1}{2}}$

if and only if the inequality

(7.10) $\varphi\left(\sum_{\lambda}b_{\lambda}x^{*}x^{\lambda}-\beta
x^{*}x\right)\leq(1-\beta)|\varphi(x)|^{2}$

holds true for every element $x\in\mathbb{C}\left[\Gamma\right]$.

###### Proof.

Let $(\pi_{\varphi},\rho_{\varphi},\mathcal{H}_{\varphi},v_{\varphi})$ be the
GNS data associated to the relative trace $\varphi$. Consider a vector
$w\in\mathcal{H}_{\varphi}$ of the form $w=\pi_{\varphi}(x)v_{\varphi}$ for
some element $x\in\mathbb{C}\left[\Gamma\right]$. We obtain the estimate

(7.11)
$\displaystyle\begin{split}\left\langle\alpha_{\varphi}(\lambda)w,w\right\rangle&=\left\langle\alpha_{\varphi}(\lambda)\pi_{\varphi}(x)v_{\varphi},\pi_{\varphi}(x)v_{\varphi}\right\rangle=\left\langle
U_{\varphi}(\lambda)\pi_{\varphi}(x)U_{\lambda}^{-1}v_{\varphi},\pi_{\varphi}(x)v_{\varphi}\right\rangle=\\\
&=\left\langle\pi_{\varphi}(x^{\lambda})v_{\varphi},\pi_{\varphi}(x)v_{\varphi}\right\rangle=\varphi\left(x^{*}x^{\lambda}\right)\end{split}$

for every element $\lambda\in\Lambda$. Note that the estimate in Equation
(7.11) is analogous to that of Equation (4.10) with the difference of using
the unitary representation $\alpha_{\varphi}$ of the group $\Lambda$ instead
of the conjugation representation $c_{\varphi}$ of the group $\Gamma$. In
particular

(7.12)
$\|\alpha_{\varphi}(a)w\|^{2}=\left\langle\alpha_{\varphi}(b)w,w\right\rangle=\sum_{\lambda\in\Lambda}\left\langle
b_{\lambda}\alpha_{\varphi}(\lambda)w,w\right\rangle=\varphi\left(\sum_{\lambda\in\Lambda}b_{\lambda}x^{*}x^{\lambda}\right).$

Once again, note that Equation (7.12) is analogous to Equation 4.11 up to
replacing the conjugation representation $c_{\varphi}$ with the representation
$\alpha_{\varphi}$. From this point onward the proof follows in exactly the
same way as that of Lemma 4.2. ∎

######  Remark 7.4.

The only property of the conjugation representation $c_{\varphi}$ and the
representation $\alpha_{\varphi}$, which was used in Lemmas 4.2 and 7.3
respectively, to express the spectral gap via formulas of traces is the fact
that the representation under consideration preserves the group algebra
$\mathbb{C}[\Gamma]$ sitting inside the Hilbert space $\mathcal{H}_{\varphi}$.

### Finite-dimensional relative characters

The correspondence taking a relative trace
$\varphi\in\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$ to the unitary
representation $U_{\varphi}:\Lambda\to\mathrm{U}(\mathcal{H}_{\varphi})$ is
continuous with respect to the pointwise convergence topology on traces and
the Fell topology on unitary representations. The proof of this fact is
mutatis mutandis the same as that of Lemma 5.6. Indeed as
$\mathrm{Inn}(\Gamma)\leq\theta(\Lambda)$ we get
$\pi_{\varphi}(\Gamma)\subseteq U_{\varphi}(\Lambda)$ so that the vector
$v_{\varphi}\in\mathcal{H}_{\varphi}$ is cyclic for the representation
$U_{\varphi}$ as well.

######  Proposition 7.5.

Let $\Lambda$ be a countable group admitting an action
$\theta:\Lambda\to\mathrm{Aut}\left(\Gamma\right)$ on the countable group
$\Gamma$. Assume that $\Lambda$ has Kazhdan’s property (T) and that
$\mathrm{Inn}(\Gamma)\leq\theta(\Lambda)$. Then any finite-dimensional
relative character $\varphi\in\mathrm{Ch}_{\Lambda}\left(\Gamma\right)$ is an
isolated point of the set $\mathrm{Ch}_{\Lambda}\left(\Gamma\right)$.

###### Proof.

Consider a sequence of relative characters
$\varphi_{n}\in\mathrm{Ch}_{\Lambda}\left(\Gamma\right)$ converging pointwise
to a finite-dimensional relative character
$\varphi\in\mathrm{Ch}_{\Lambda}\left(\Gamma\right)$. As
$\mathrm{Inn}(\Gamma)\leq\theta(\Lambda)$ and taking into account Proposition
7.1 we get $N_{\varphi}^{\prime}=U_{\varphi}(\Lambda)^{\prime}$. On the other
hand, since $\varphi$ is a relative character we have
$N_{\varphi}^{\prime}=\mathbb{C}\cdot\mathrm{Id}$. Therefore
$U_{\varphi}(\Lambda)^{\prime}=\mathbb{C}\cdot\mathrm{Id}$ which means that
$U_{\varphi}(\Lambda)$ is an irreducible representation. The same argument
applied with respect to each relative character $\varphi_{n}$ shows that each
representation $U_{\varphi_{n}}$ is irreducible.

The paragraph preceding this proposition shows that the unitary
representations corresponding to the relative characters $\varphi_{n}$ and
$\varphi$ satisfy $U_{\varphi_{n}}\to U_{\varphi}$ in the Fell topology. The
assumption that the relative character $\varphi$ is finite-dimensional means
that the Hilbert space $\mathcal{H}_{\varphi}$ is finite-dimensional. In
particular, the representation $U_{\varphi}$ is finite-dimensional. Recall
that for Kazhdan groups, finite-dimensional representations are isolated
points of the unitary dual in the Fell topology [BdLHV08, Theorem 1.2.5]. It
follows that $U_{\varphi_{n}}=U_{\varphi}$ for all $n\in\mathbb{N}$
sufficiently large. In particular $\pi_{\varphi_{n}}=\pi_{\varphi}$ so that
$\varphi_{n}=\varphi$ for all $n\in\mathbb{N}$ sufficiently large. ∎

### The simplex of relative traces

Assume that the group $\Lambda$ has property (T) and that
$\mathrm{Inn}(\Gamma)\leq\theta(\Lambda)$. The fact that the space of relative
traces $\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$ is a Bauer simplex follows
along the same lines as the proof of Theorem 1.6. Let us provide a bit more
detail.

###### Proof of Theorem 1.9.

Consider a sequence of relative characters
$\varphi_{n}\in\mathrm{Ch}_{\Lambda}\left(\Gamma\right)$ converging pointwise
to some relative trace $\varphi\in\mathrm{Tr}_{\Lambda}\left(\Gamma\right)$.
For each $n\in\mathbb{N}$ the subspace of the Hilbert space
$\mathcal{H}_{\varphi_{n}}$ consisting of ${\alpha_{\varphi_{n}}}$-invariant
vectors is one-dimensional and is spanned by the cyclic vector
$v_{\varphi_{n}}$, see Proposition 7.2. Since the group $\Lambda$ has
Kazhdan’s property (T) and according to Lemma 7.3 there is some constant
$0<\beta<1$ (depending on the chosen element $a\in\ell^{1}(\Lambda)$) such
that Equation (7.10) holds true with respect to all the relative characters
$\varphi_{n}$. The validity of this equation passes to the limiting trace
$\varphi$ with the same value of $\beta$. Therefore the relative trace
$\varphi$ is a relative character by the converse direction of Lemma 7.3 and
of Proposition 7.2. The second part of the statement concerning finite-
dimensional relative characters has already been established in Proposition
7.5. ∎

The last part of our work deals with relative characters of the abelian group
$\mathbb{Z}^{d}$.

######  Corollary 7.6.

Assume that $d\geq 3$. Any sequence of pairwise distinct relative characters
$\varphi_{n}\in\mathrm{Ch}_{\mathrm{SL}_{d}(\mathbb{Z})}\left(\mathbb{Z}^{d}\right)$
converges pointwise to the Dirac trace $\delta_{e}$ on $\mathbb{Z}^{d}$.

The Pontryagin dual of the free abelian group $\mathbb{Z}^{d}$ is the torus
$\mathbb{T}^{d}$. Bochner’s theorem sets up an
$\mathrm{SL}_{d}(\mathbb{Z})$-equivariant affine homeomorphism
$\mathrm{Tr}(\mathbb{Z}^{d})\cong\mathrm{Prob}(\mathbb{T}^{d})$ of compact
convex sets, where the space of probability measures
$\mathrm{Prob}(\mathbb{T}^{d})$ is equipped with the weak-$*$ topology.
Therefore Corollary 7.6 is equivalent to the following dynamical statement:
_Let $\mu_{n}$ be any sequence of pairwise distinct ergodic
$\mathrm{SL}_{d}(\mathbb{Z})$-invariant Borel probability measures on the
torus $\mathbb{T}^{d}$. Then the sequence $\mu_{n}$ converges in the weak-$*$
topology to the Haar measure on the torus $\mathbb{T}^{d}$._ This statement is
certainly well known to experts and is valid for all $d\geq 2$ (however we
could not locate a reference in the literature).

###### Proof of Corollary 7.6.

Let
$\varphi_{n}\in\mathrm{Ch}_{\mathrm{SL}_{d}(\mathbb{Z})}\left(\mathbb{Z}^{d}\right)$
be any sequence of pairwise distinct relative characters. Consider an
accumulation point
$\varphi\in\mathrm{Tr}_{\mathrm{SL}_{d}(\mathbb{Z})}\left(\mathbb{Z}^{d}\right)$
so that $\varphi$ is a relative trace. Theorem 1.9 says that $\varphi$ is in
fact an infinite-dimensional relative character. In particular, the
corresponding ergodic $\mathrm{SL}_{d}(\mathbb{Z})$-invariant Borel
probability measure $\widehat{\varphi}\in\mathrm{Prob}(\mathbb{T}^{d})$ is
non-atomic. However the only non-atomic
$\mathrm{SL}_{d}(\mathbb{Z})$-invariant Borel probability measures on the
torus $\mathbb{T}^{d}$ is the Haar measure [Bur91, Proposition 9]. In other
words, the character $\varphi$ must be the Dirac trace $\delta_{e}$. ∎

We conclude with a vanishing result for the semidirect product
$\mathrm{Aut}(\mathbb{Z}^{d})\ltimes\mathbb{Z}^{d}$.

######  Theorem 7.7.

Let
$\varphi_{n}\in\mathrm{Ch}\left(\mathrm{GL}_{d}(\mathbb{Z})\ltimes\mathbb{Z}^{d}\right)$
be any sequence of characters such that the restrictions
$\varphi_{n\,|\mathbb{Z}^{d}}$ are pairwise distinct. Then the characters
$\varphi_{n}$ converge pointwise to the Dirac character
$\delta_{e}\in\mathrm{Ch}\left(\mathrm{GL}_{d}(\mathbb{Z})\ltimes\mathbb{Z}^{d}\right)$.

###### Proof.

Assume that $d\geq 3$. Consider the semidirect product
$\Gamma=\mathrm{GL}_{d}(\mathbb{Z})\ltimes\mathbb{Z}^{d}$. Let
$\varphi_{n}\in\mathrm{Ch}\left(\Gamma\right)$ be any sequence of characters
whose restrictions to the subgroup $\mathbb{Z}^{d}$ are pairwise distinct. Let
$\varphi\in\mathrm{Tr}\left(\Gamma\right)$ be any accumulation point of the
sequence $\varphi_{n}$. We know that $\varphi(\gamma)=0$ for any element
$\gamma\in\Gamma\setminus\mathrm{Rad}\left(\Gamma\right)$ where
$\mathrm{Rad}\left(\Gamma\right)$ is the amenable radical given by
$\mathrm{Rad}\left(\Gamma\right)=\mathrm{Z}(\mathrm{GL}_{d}(\mathbb{Z}))\ltimes\mathbb{Z}^{d}$,
see Theorem 6.6. In addition, the fact that $\varphi(\gamma)=0$ for any
element $\gamma\in\mathbb{Z}^{d}\setminus\\{e\\}$ follows from Corollary 7.6.

It remains to show that $\varphi(\gamma)=0$ for all elements
$\gamma\in\mathrm{Rad}\left(\Gamma\right)\setminus\mathbb{Z}^{d}$. Note that
any given element
$\gamma\in\mathrm{Rad}\left(\Gamma\right)\setminus\mathbb{Z}^{d}$ is of the
form $\gamma=(-\mathrm{Id},u)$ for some $u\in\mathbb{Z}^{d}$. Take any
sequence of pairwise distinct elements $v_{n}\in\mathbb{Z}^{d}$. The
commutators given by

(7.13)
$\left[\gamma,v_{n}\right]=\left[(-\mathrm{Id},u),(\mathrm{Id},v_{n})\right]=(\mathrm{Id},2(v_{n}-u))$

are pairwise distinct as well. Since the trace $\varphi$ vanishes on
$\mathbb{Z}^{d}\setminus\\{e\\}$ we conclude that $\varphi(\gamma)=0$ using
[LL23, Lemma 4.13]. ∎

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