# The Bohr compactification of an arithmetic group

Bachir Bekka

###### Abstract.

Given a group $\Gamma,$ its Bohr compactification
$\operatorname{Bohr}(\Gamma)$ and its profinite completion
$\operatorname{Prof}(\Gamma)$ are compact groups naturally associated to
$\Gamma$; moreover, $\operatorname{Prof}(\Gamma)$ can be identified with the
quotient of $\operatorname{Bohr}(\Gamma)$ by its connected component
$\operatorname{Bohr}(\Gamma)_{0}.$ We study the structure of
$\operatorname{Bohr}(\Gamma)$ for an arithmetic subgroup $\Gamma$ of an
algebraic group $\mathbf{G}$ over $\mathbf{Q}$. When $\mathbf{G}$ is
unipotent, we show that $\operatorname{Bohr}(\Gamma)$ can be identified with
the direct product $\operatorname{Bohr}(\Gamma^{\rm
Ab})_{0}\times\operatorname{Prof}(\Gamma)$, where $\Gamma^{\rm
Ab}=\Gamma/[\Gamma,\Gamma]$ is the abelianization of $\Gamma.$ In the general
case, using a Levi decomposition $\mathbf{G}=\mathbf{U}\rtimes\mathbf{H}$
(where $\mathbf{U}$ is unipotent and $\mathbf{H}$ is reductive), we show that
$\operatorname{Bohr}(\Gamma)$ can be described as the semi-direct product of a
certain quotient of $\operatorname{Bohr}(\Gamma\cap\mathbf{U})$ with
$\operatorname{Bohr}(\Gamma\cap\mathbf{H})$. When $\mathbf{G}$ is simple and
has higher ${\mathbf{R}}$-rank, $\operatorname{Bohr}(\Gamma)$ is isomorphic,
up to a finite group, to the product $K\times\operatorname{Prof}(\Gamma),$
where $K$ is the maximal compact factor of $\mathbf{G}({\mathbf{R}}).$

###### 2000 Mathematics Subject Classification:

22D10; 22C05; 20E18

The author acknowledges the support by the ANR (French Agence Nationale de la
Recherche) through the project Labex Lebesgue (ANR-11-LABX-0020-01) .

##  1\. Introduction

Given a topological group $G,$ the Bohr compactification of $G$ is a pair
$(\operatorname{Bohr}(G),\beta)$ consisting of a compact (Hausdorff) group
$\operatorname{Bohr}(G)$ and a continuous homomorphism
$\beta:G\to\operatorname{Bohr}(G)$ with dense image, satisfying the following
universal property: for every compact group $K$ and every continuous
homomorphism $\alpha\,\colon G\to K$, there exists a continuous homomorphism
$\alpha^{\prime}\,\colon\operatorname{Bohr}(G)\to K$ such that the diagram

${\operatorname{Bohr}(G)}$${G}$${K}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$

commutes. The pair $(\operatorname{Bohr}(G),\beta)$ is unique in the following
sense: if $(K^{\prime},\beta^{\prime})$ is a pair consisting of a compact
group $K^{\prime}$ and a continuous homomorphism $\beta^{\prime}:G\to
K^{\prime}$ with dense image satisfying the same universal property (such a
pair will be called a Bohr compactification of $G$), then there exists an
isomorphism $\alpha:\operatorname{Bohr}(G)\to K^{\prime}$ of topological
groups such that $\beta^{\prime}=\alpha\circ\beta$.

The compact group $\operatorname{Bohr}(G)$ was first introduced by A. Weil
([Weil, Chap.VII]) as a tool for the study of almost periodic functions on
$G$, a subject initiated by H. Bohr ([Bohr1], [Bohr2]) in the case
$G={\mathbf{R}}$ and generalized to other groups by J. von Neumann ([VN])
among others. For more on this subject, see [Dixm--C*, §16] or [BH, 4.C]).

The group $\operatorname{Bohr}(\Gamma)$ has been determined for only very few
non abelian _discrete_ groups $\Gamma$ (for some general results, see [Hart-
Kunen] and [Holm]; for the well-known case of abelian groups, see [Anzai-
Kakutani1] and Section 11).

In contrast, there is a second much more studied completion of $\Gamma$,
namely the profinite completion of $\Gamma$, which is a pair
$(\operatorname{Prof}(\Gamma),\alpha)$ consisting of a profinite group (that
is, a projective limit of finite groups) $\operatorname{Prof}(\Gamma)$
satisfying a similar universal property with respect to such groups, together
with a homomorphism with $\alpha\,\colon\Gamma\to\operatorname{Prof}(\Gamma)$
with dense image. The group $\operatorname{Prof}(\Gamma)$ can be realized as
the projective limit $\varprojlim\Gamma/H$, where $H$ runs over the family of
the normal subgroups of finite index of $\Gamma.$ For all this, see [Ribes-
Zal].

The universal property of $\operatorname{Bohr}(\Gamma)$ gives rise to a
continuous epimorphism
$\alpha^{\prime}\,\colon\operatorname{Bohr}(\Gamma)\to\operatorname{Prof}(\Gamma).$
It is easy to see (see Proposition 7 below) that the kernel of
$\alpha^{\prime}$ is $\operatorname{Bohr}(\Gamma)_{0}$, the connected
component of $\operatorname{Bohr}(\Gamma);$ so, we have a short exact sequence

${1}$${\operatorname{Bohr}(\Gamma)_{0}}$${\operatorname{Bohr}(\Gamma)}$${\operatorname{Prof}(\Gamma)}$${1.}$

In this paper, we will deal with the case where $\Gamma$ is an arithmetic
subgroup in a linear algebraic group. The setting is as follows. Let
$\mathbf{G}$ be a connected linear algebraic group over $\mathbf{Q}$ with a
fixed faithful representation $\rho:\mathbf{G}\to GL_{m}.$ We consider the
subgroup $\mathbf{G}({\mathbf{Z}})$ of the group $\mathbf{G}(\mathbf{Q})$ of
$\mathbf{Q}$-points of $\mathbf{G}$, that is,

$\mathbf{G}({\mathbf{Z}})=\rho^{-1}\left(\rho(\mathbf{G})\cap
GL_{m}({\mathbf{Z}})\right).$

A subgroup $\Gamma$ of $\mathbf{G}(\mathbf{Q})$ is called an arithmetic
subgroup if $\Gamma$ is commensurable to $\mathbf{G}({\mathbf{Z}})$, that is,
$\Gamma\cap\mathbf{G}({\mathbf{Z}})$ has finite index in both $\Gamma$ and
$\mathbf{G}({\mathbf{Z}})$. Observe that $\Gamma$ is a discrete subgroup of
the real Lie group $\mathbf{G}({\mathbf{R}}).$

We first deal with the case where $\mathbf{G}$ is unipotent. More generally,
we describe the Bohr compactification of any finitely generated nilpotent
group. Observe that an arithmetic subgroup in a unipotent algebraic
$\mathbf{Q}$-group is finitely generated (see Corollary 2 of Theorem 2.10 in
[Raghunathan]).

For two topological groups $H$ and $L,$ we write $H\cong L$ if $H$ and $L$ are
topologically isomorphic. We observe that, when $\Delta$ is a finitely
generated abelian group, $\operatorname{Bohr}(\Delta)$ splits as a direct sum
$\operatorname{Bohr}(\Delta)=\operatorname{Bohr}(\Delta)_{0}\oplus\operatorname{Prof}(\Delta)$;
see Proposition 11.

######  Theorem 1.

Let $\Gamma$ be a finitely generated nilpotent group. We have a direct product
decomposition

$\operatorname{Bohr}(\Gamma)\cong\operatorname{Bohr}(\Gamma^{\rm
Ab})_{0}\times\operatorname{Prof}(\Gamma),$

where $\Gamma^{\rm Ab}=\Gamma/[\Gamma,\Gamma]$ is the abelianization of
$\Gamma.$ This isomorphism is induced by the natural maps
$\Gamma\to\operatorname{Bohr}(\Gamma^{\rm Ab})$ and
$\Gamma\to\operatorname{Prof}(\Gamma),$ together with the projection
$\operatorname{Bohr}(\Gamma^{\rm Ab})\to\operatorname{Bohr}(\Gamma^{\rm
Ab})_{0}$.

A crucial tool in the proof of Theorem 1 is the fact that elements in the
commutator subgroup $[\Gamma,\Gamma]$ of a nilpotent group $\Gamma$ are
distorted (see Proposition 15).

We now turn to the case of a general algebraic group $\mathbf{G}$ over
$\mathbf{Q}.$ Let $\mathbf{U}$ be the unipotent radical of $\mathbf{G}$. Then
$\mathbf{U}$ is defined over $\mathbf{Q}$ and there exists a connected
reductive $\mathbf{Q}$-subgroup $\mathbf{H}$ such that we have a Levi
decomposition as semi-direct product $\mathbf{G}=\mathbf{U}\rtimes\mathbf{H}$
(see [Mostow]).

The group $\Lambda=\mathbf{H}({\mathbf{Z}})$ acts by automorphisms on
$\Delta=\mathbf{U}({\mathbf{Z}})$ and hence on $\operatorname{Bohr}(\Delta),$
by the universal property of $\operatorname{Bohr}(\Delta).$ In general, this
action does not extend to an action of $\operatorname{Bohr}(\Lambda)$ on
$\operatorname{Bohr}(\Delta).$ However, as we will see below (proof of Theorem
2), $\operatorname{Bohr}(\Lambda)$ acts naturally by automorphisms on an
appropriate quotient of $\operatorname{Bohr}(\Delta).$

Observe that (see [Borel-HC, Corollary 4.6]) every arithmetic subgroup of
$\mathbf{G}(\mathbf{Q})$ is commensurable to
$\Delta({\mathbf{Z}})\rtimes\mathbf{H}({\mathbf{Z}}).$ Recall that two
topological groups $G_{1}$ and $G_{2}$ are (abstractly) commensurable if there
exist finite index subgroups $H_{1}$ and $H_{2}$ of $G_{1}$ and $G_{2}$ such
that $H_{1}$ is topologically isomorphic to $H_{1}.$ If this is the case, then
$\operatorname{Bohr}(G_{1})$ and $\operatorname{Bohr}(G_{2})$ are
commensurable; in fact, each one of the groups $\operatorname{Bohr}(G_{1})$ or
$\operatorname{Bohr}(G_{2})$ can be described in terms of the other (see
Propositions 8 and 9 ). For this reason, we will often deal with only one
chosen representative of the commensurability class of an arithmetic group.

######  Theorem 2.

Let $\mathbf{G}$ be a connected linear algebraic group over $\mathbf{Q},$ with
Levi decomposition $\mathbf{G}=\mathbf{U}\rtimes\mathbf{H}.$ Set
$\Lambda:=\mathbf{H}({\mathbf{Z}}),\Delta:=\mathbf{U}({\mathbf{Z}}),$ and
$\Gamma:=\Delta\rtimes\Lambda.$ Let $\widehat{\Delta^{\rm
Ab}}_{\Lambda-\rm{fin}}$ be the subgroup of the dual group
$\widehat{\Delta^{\rm Ab}}$ of $\Delta^{\rm Ab}$ consisting of the characters
with finite $\Lambda$-orbit. We have a semi-direct decomposition

$\operatorname{Bohr}(\Gamma)\cong(Q\times\operatorname{Prof}(\Delta))\rtimes\operatorname{Bohr}(\Lambda),$

where $Q$ is the connected component of $\operatorname{Bohr}(\Delta^{\rm
Ab})/N$ and $N$ is the annihilator of $\widehat{\Delta^{\rm
Ab}}_{\Lambda-\rm{fin}}$ in $\operatorname{Bohr}(\Delta^{\rm Ab})$. This
isomorphism is induced by the natural homomophisms
$\Delta\to\operatorname{Bohr}(\Delta^{\rm Ab})/N$ and
$\Lambda\to\operatorname{Bohr}(\Lambda).$

Theorems 1 and 2 reduce the determination of $\operatorname{Bohr}(\Gamma)$ for
an arithmetic group $\Gamma$ in $\mathbf{G}$ to the case where $\mathbf{G}$ is
reductive. We have a further reduction to the case where $\mathbf{G}$ is
simply connected and almost simple. Indeed, recall that a group $L$ is the
almost direct product of subgroups $L_{1},\dots,L_{n}$ if the product map
$L_{1}\times\cdots\times L_{n}\to L$ is a surjective homomorphism with finite
kernel.

Let $\mathbf{G}$ be a connected reductive algebraic group over $\mathbf{Q}.$
The commutator subgroup $\mathbf{L}:=[\mathbf{G},\mathbf{G}]$ of $\mathbf{G}$
is a connected semi-simple $\mathbf{Q}$-group and $\mathbf{G}$ is an almost
direct product $\mathbf{G}=\mathbf{T}\mathbf{L}$ for a central
$\mathbf{Q}$-torus $\mathbf{T}$ (see (14.2) and (18.2) in [Borel-Book])
Moreover, $\mathbf{L}$ is an almost direct product
$\mathbf{L}=\mathbf{L}_{1}\cdots\mathbf{L}_{n}$ of connected almost
$\mathbf{Q}$-simple $\mathbf{Q}$-subgroups $\mathbf{L}_{i},$ called the almost
$\mathbf{Q}$-simple factors of $\mathbf{L}$ (see [Borel-Book, (22.10)]). For
every $i\in\\{1,\dots,n\\},$ let $\widetilde{\mathbf{L}}_{i}$ be the simply
connected covering group $\mathbf{L}_{i}.$ Set
$\widetilde{\mathbf{G}}=\mathbf{T}\times\widetilde{\mathbf{L}_{1}}\times\cdots\times\widetilde{\mathbf{L}_{n}}.$
Let $\widetilde{\Gamma}$ be the arithmetic subgroup
$\mathbf{T}({\mathbf{Z}})\times\widetilde{\mathbf{L}_{1}}({\mathbf{Z}})\times\cdots\times\widetilde{\mathbf{L}_{n}}({\mathbf{Z}})$
in $\widetilde{\mathbf{G}}(\mathbf{Q}).$ The image $\Gamma$ of
$\widetilde{\Gamma}$ under the isogeny $p:\widetilde{\mathbf{G}}\to\mathbf{G}$
is an arithmetic subgroup of $\mathbf{G}(\mathbf{Q})$ (see Corollaries 6.4 and
6.11 in [Borel-HC]). The map $p:\widetilde{\Gamma}\to\Gamma$ induces an
isomorphism
$\operatorname{Bohr}(\Gamma)\cong\operatorname{Bohr}(\widetilde{\Gamma})/F$,
where $F$ is the finite normal subgroup $F=\widetilde{\beta}(\ker p)$ and
$\widetilde{\beta}:\widetilde{\Gamma}\to\operatorname{Bohr}(\widetilde{\Gamma})$
is the natural map (see Proposition 10).

As an easy consequence of Margulis’ superrigidity results, we give a
description of the Bohr compactification of an arithmetic lattice in a simple
algebraic $\mathbf{Q}$-group $\mathbf{G}$ under a higher rank assumption. Such
a description does not seem possible for arbitrary $\mathbf{G}$. For instance,
the free non abelian group $F_{2}$ on two generators is an arithmetic lattice
in $SL_{2}(\mathbf{Q})$, but we know of no simple description of
$\operatorname{Bohr}(F_{2}).$

######  Theorem 3.

Let $\mathbf{G}$ be a connected, simply connected, and almost simple
$\mathbf{Q}$-group. Assume that the real semisimple Lie group
$\mathbf{G}({\mathbf{R}})$ is not locally isomorphic to any group of the form
$SO(m,1)\times K$ or $SU(m,1)\times K$ for a compact Lie group $K$. Let
$\mathbf{G}_{\rm nc}$ be the product of the almost ${\mathbf{R}}$-simple
factors $\mathbf{G}_{i}$ of $\mathbf{G}$ for which
$\mathbf{G}_{i}({\mathbf{R}})$ is non compact. Let
$\Gamma\subset\mathbf{G}(\mathbf{Q})$ be an arithmetic subgroup. We have a
direct product decomposition

$\operatorname{Bohr}(\Gamma)\cong\operatorname{Bohr}(\Gamma)_{0}\times\operatorname{Prof}(\Gamma)$

and an isomorphism

$\operatorname{Bohr}(\Gamma)_{0}\cong\mathbf{G}({\mathbf{R}})/\mathbf{G}_{\rm
nc}({\mathbf{R}}),$

induced by the natural maps
$\Gamma\to\mathbf{G}({\mathbf{R}})/\mathbf{G}({\mathbf{R}})_{\rm nc}$ and
$\Gamma\to\operatorname{Prof}(\Gamma)$.

A group $\Gamma$ as in Theorem 3 is an irreducible lattice in the Lie group
$G=\mathbf{G}({\mathbf{R}})$, that is, the homogeneous space $G/\Gamma$
carries a $G$-invariant probability measure; moreover, $\Gamma$ is cocompact
in $G$ if and only if $\mathbf{G}$ is anisotropic over $\mathbf{Q}$ (for all
this, see [Borel-HC, (7.8), (11.6)]). The following corollary is a direct
consequence of Theorem 3 and of the fact that a non cocompact arithmetic
lattice in a semisimple Lie group has nontrivial unipotent elements (see
[Witte, (5.5.14)]).

######  Corollary 4.

With the notation as in Theorem 3, assume that $\mathbf{G}$ is isotropic over
$\mathbf{Q}.$ For every arithmetic subgroup $\Gamma$ of
$\mathbf{G}(\mathbf{Q}),$ the natural map
$\operatorname{Bohr}(\Gamma)\to\operatorname{Prof}(\Gamma)$ is an isomorphism.

As shown in Section 6, it may happen that
$\operatorname{Bohr}(\mathbf{G}({\mathbf{Z}}))\cong\operatorname{Prof}(\mathbf{G}({\mathbf{Z}})),$
even when $\mathbf{G}({\mathbf{Z}})$ is cocompact in
$\mathbf{G}({\mathbf{R}}).$.

A general arithmetic lattice $\Gamma$ has a third completion: the congruence
completion $\operatorname{Cong}(\Gamma)$ of $\Gamma$ is the projective limit
$\varprojlim\Gamma/H$, where $H$ runs over the family of the congruence
subgroups of $\Gamma$; recall that a normal subgroup of $\Gamma$ is a
congruence subgroup if it contains the kernel of the map
$\mathbf{G}({\mathbf{Z}})\to\mathbf{G}({\mathbf{Z}}/N{\mathbf{Z}})$ of the
reduction modulo $N,$ for some integer $N\geq 1.$ There is a natural
surjective homomorphism
$\pi:\operatorname{Prof}(\Gamma)\to\operatorname{Cong}(\Gamma)$. The so-called
congruence subgroup problem asks whether $\pi$ is injective and hence an
isomorphism of topological groups; more generally, one can ask for a
description of the kernel of $\pi.$ This problem has been extensively studied
for arithmetic subgroups (and, more generally, for $S$-arithmetic subgroups)
in various algebraic groups; for instance, it is known that $\pi$ is an
isomorphism when $\Gamma=SL_{n}({\mathbf{Z}})$ for $n\geq 3$ or
$\Gamma=Sp_{2n}({\mathbf{Z}})$ for $n\geq 2$ (see [BMS]); moreover, the same
conclusion is true when $\Gamma=\mathbf{T}({\mathbf{Z}})$ for a torus
$\mathbf{T}$ (see [Chevalley]) and when $\Gamma=\mathbf{U}({\mathbf{Z}})$ for
a unipotent group $\mathbf{U}$ (see Proposition 16 below). For more on the
congruence subgroup problem, see for instance [Raghunathan-CSP] or [Platonov,
§9.5].

This paper is organized as follows. In Section 2, we establish some general
facts about the Bohr compactifications of commensurable groups and the
relationship between Bohr compactifications and unitary representations; we
also give an explicit description of the Bohr compactification for a finitely
generated abelian group. In Section 3, we give the proof of Theorem 1. Section
4 contains the proof of Theorem 2 and Section 5 the proof of Theorem 3.
Section 6 is devoted to the explicit computation of the Bohr compactification
for various examples of arithmetic groups.

##  2\. Some preliminaries

###  2.1. Bohr compactifications and unitary representations

Given a topological group $G,$ we will consider finite dimensional unitary
representations of $G,$ that is, continuous homomorphisms $G\to U(n)$. Two
such representations are equivalent if they are conjugate by a unitary matrix.
A representation $\pi$ is irreducible if ${\mathbf{C}}^{n}$ and $\\{0\\}$
there are only $\pi(G)$-invariant subspaces of ${\mathbf{C}}^{n}$. We denote
by $\operatorname{Rep}_{\rm fd}(G)$ the set of equivalence classes of finite
dimensional unitary representations of $G$ and by $\widehat{G}_{\rm fd}$ the
subset of irreducible ones. Every $\pi\in\operatorname{Rep}_{\rm fd}(G)$ is a
direct sum of representations from $\widehat{G}_{\rm fd}$

When $K$ is a compact group, every irreducible unitary representation of $K$
is finite dimensional and $\widehat{K}_{\rm fd}=\widehat{K}$ is the unitary
dual space of $K.$ By the Peter-Weyl theorem, $\widehat{K}$ separates the
points of $K$.

Let $\beta:G\to H$ be a a continuous homomorphism of topological groups $G$
and $H$ with dense image; then $\beta$ induces _injective_ maps

$\widehat{\beta}:\operatorname{Rep}_{\rm fd}(H)\to\operatorname{Rep}_{\rm
fd}(G)\qquad\text{and}\qquad\widehat{\beta}:\widehat{H}_{\rm
fd}\to\widehat{G}_{\rm fd},$

given by $\widehat{\beta}(\pi)=\pi\circ\beta$ for
$\pi\in\operatorname{Rep}_{\rm fd}(H).$ The following proposition, which may
be considered as well-known, is a useful tool for identifying the Bohr
compactification of a group.

######  Proposition 5.

Let $G$ be a topological group, $K$ a compact group, and $\beta:G\to K$ a
continuous homomorphism with dense image. The following properties are
equivalent:

  * (i)

$(K,\beta)$ is a Bohr compactification of $G;$

  * (ii)

the induced map $\widehat{\beta}:\widehat{K}\to\widehat{G}_{\rm fd}$ is
surjective;

  * (iii)

the induced map $\widehat{\beta}:\operatorname{Rep}_{\rm
fd}(K)\to\operatorname{Rep}_{\rm fd}(G)$ is surjective.

###### Proof.

Assume that (i) holds and let $\pi:G\to U(n)$ be an irreducible representation
of $G$; by the universal property of the Bohr compactification, there exists a
continuous homomorphism $\pi^{\prime}:K\to U(n)$ such that
$\pi=\widehat{\beta}(\pi^{\prime})$ and (ii) follows.

Conversely, assume that (ii) holds. Let $L$ be a compact group and
$\alpha:G\to L$ a continuous homomorphism with dense image. Choose a family
$\pi_{i}:L\to U(n_{i})$ of representatives of $\widehat{L}.$ By the Peter-Weyl
theorem, we may identify $L$ with its image in $\prod_{i}U(n_{i})$ under the
map $x\mapsto\oplus_{i}\pi_{i}(x)$ For every $i$, we have
$\pi_{i}\circ\alpha\in\widehat{G}_{\rm fd}$ and hence
$\pi_{i}\circ\alpha=\widehat{\beta}(\pi^{\prime}_{i})=\pi_{i}^{\prime}\circ\beta$
for some representation $\pi^{\prime}_{i}:K\to U(n_{i})$ of $K$. Define a
continuous homomorphism

$\alpha^{\prime}:K\to\prod_{i}U(n_{i})\qquad
x\mapsto\oplus_{i}\pi_{i}^{\prime}(x).$

We have $\alpha^{\prime}\circ\beta=\alpha$ and hence

$\alpha^{\prime}(K)=\alpha^{\prime}\left(\overline{\beta(G)}\right)\subset\overline{\alpha(G)}=L.$

So, (i) and (ii) are equivalent. It is obvious that (ii) is equivalent to
(iii). ∎

The profinite completion $(\operatorname{Prof}(G),\alpha)$ of $G$ may be
similarly characterized in terms of certain unitary representations of $G.$
Recall first that $(\operatorname{Prof}(G),\alpha)$ is a pair consisting of a
profinite group $\operatorname{Prof}(G)$ and a continuous homomorphism
$\alpha:G\to\operatorname{Prof}(G)$ with dense image, satisfying the following
universal property: for every profinite group $K$ and every continuous
homomorphism $f\,\colon G\to K$, there exists a continuous homomorphism
$f^{\prime}\,\colon\operatorname{Bohr}(G)\to K$ such that the diagram

${\operatorname{Prof}(G)}$${G}$${K}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\alpha}$$\scriptstyle{f}$

commutes. Recall that the class of profinite groups coincides with the class
of totally disconnected compact groups (see [BH, Proposition 4.C.10]).

Denote by $\operatorname{Rep}_{\rm finite}(G)$ the set of equivalence classes
of finite dimensional unitary representations $\pi$ of $G$ for which $\pi(G)$
is finite; let $\widehat{G}_{\rm finite}$ be the subset of irreducible
representations from $\operatorname{Rep}_{\rm finite}(G)$.

If $\alpha:G\to H$ is a continuous homomorphism of topological groups $G$ and
$H$ with dense image, then $\beta$ induces _injective_ maps

$\widehat{\alpha}:\operatorname{Rep}_{\rm finite}(H)\to\operatorname{Rep}_{\rm
finite}(G)\qquad\text{and}\qquad\widehat{\alpha}:\widehat{H}_{\rm
finite}\to\widehat{G}_{\rm finite}.$

Observe that $\widehat{K}=\widehat{K}_{\rm finite}$ if $K$ is a profinite
group. (Conversely, it follows from Peter-Weyl theorem that, if $K$ is a
compact group with $\widehat{K}=\widehat{K}_{\rm finite}$, then $K$ is
profinite.) The proof of the following proposition is similar to the proof of
Proposition 5 and will be omitted.

######  Proposition 6.

Let $K$ be a totally disconnected compact group and $\alpha:G\to K$ a
continuous homomorphism with dense image. The following properties are
equivalent:

  * (i)

$(K,\alpha)$ is a profinite completion of $G;$

  * (ii)

the induced map $\widehat{\alpha}:\widehat{K}\to\widehat{G}_{\rm finite}$ is
surjective;

  * (ii)

the induced map $\widehat{\beta}:\operatorname{Rep}_{\rm
finite}(K)\to\operatorname{Rep}_{\rm finite}(G)$ is surjective.

The universal property of $\operatorname{Bohr}(G)$ implies that there is a
continuous epimorphism
$\alpha^{\prime}\,\colon\operatorname{Bohr}(G)\to\operatorname{Prof}(G)$ such
that the diagram

${\operatorname{Bohr}(G)}$${G}$${\operatorname{Prof}(G)}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$

commutes. We record the following elementary but basic fact mentioned in the
introduction.

######  Proposition 7.

The kernel of
$\alpha^{\prime}:\operatorname{Bohr}(G)\to\operatorname{Prof}(G)$ coincides
with the connected component $\operatorname{Bohr}(G)_{0}$ of
$\operatorname{Bohr}(G).$

###### Proof.

Since $\operatorname{Bohr}(G)_{0}$ is connected and $\operatorname{Prof}(G)$
is totally disconnected, $\operatorname{Bohr}(G)_{0}$ is contained in ${\rm
Ker}\alpha^{\prime}$. So, $\alpha^{\prime}$ factorizes to a continuous
epimorphism $\alpha^{\prime\prime}:K\to\operatorname{Prof}(G),$ where
$K:=\operatorname{Bohr}(G)/\operatorname{Bohr}(G)_{0}$ and we have a
commutative diagram

${K}$${G}$${\operatorname{Prof}(G).}$$\scriptstyle{\alpha^{\prime\prime}}$$\scriptstyle{p\circ\beta}$$\scriptstyle{\alpha}$

where $p:\operatorname{Bohr}(G)\to K$ is the canonical epimorphism. Since $K$
is a totally disconnected compact group, there exists a continuous epimorphism
$f:\operatorname{Prof}(G)\to K$ and we have a commutative diagram

${K}$${G}$${\operatorname{Prof}(G).}$$\scriptstyle{p\circ\beta}$$\scriptstyle{\alpha}$$\scriptstyle{f}$

For every $g\in G,$ we have

$f(\alpha^{\prime\prime}(p\circ\beta(g)))=f(\alpha(g))=p\circ\beta(g);$

since $p\circ\beta(G)$ is dense in $K,$ it follows that
$f\circ\alpha^{\prime\prime}$ is the identity on $K.$ This implies that
$\alpha^{\prime\prime}$ is injective and hence an isomorphism. ∎

###  2.2. Bohr compactifications of commensurable groups

Let $G$ be a topological group and $H$ be a closed subgroup of finite index in
$G$. We first determine $\operatorname{Bohr}(H)$ in terms of
$\operatorname{Bohr}(G)$.

######  Proposition 8.

Let $(\operatorname{Bohr}(G),\beta)$ be the Bohr compactification of $G$. Set
$K:=\overline{\beta(H)}.$

  * (i)

$K$ is a subgroup of finite index of $\operatorname{Bohr}(G)$.

  * (ii)

$(K,\beta|_{H})$ is a Bohr compactification of $H.$

  * (iii)

$K$ and $\operatorname{Bohr}(G)$ have the same connected component of the
identity.

###### Proof.

Item (i) is obvious and Item (iii) follows from Item (i). To show Item (ii),
let $\pi$ be a unitary representation of $H$ on ${\mathbf{C}}^{n}.$ Since $H$
has finite index in $H,$ the induced representation
$\rho:=\operatorname{Ind}_{H}^{G}\pi$, which is a unitary representation of
$G,$ is finite dimensional. Hence, there exists
$\rho^{\prime}\in\operatorname{Rep}_{\rm fd}(\operatorname{Bohr}(G))$ such
that $\rho=\rho^{\prime}\circ\beta$. Now, $\pi$ is equivalent to a
subrepresentation of the restriction of $\rho$ to $H$ (see [BH, 1.F]); so, we
may identify $\pi$ with the representation of $H$ defined by a
$\rho(H)$-invariant subspace $W$ of the space of $\rho.$ Then $W$ is
$\rho^{\prime}(K)$-invariant and defines therefore a representation
$\pi^{\prime}$ of $K.$ We have $\pi=\pi^{\prime}\circ(\beta|_{H})$ and
Proposition 5 shows that Item (ii) holds. ∎

Next, we want to determine $\operatorname{Bohr}(G)$ in terms of
$\operatorname{Bohr}(H)$.

Given a compact group $K$ and a finite set $X,$ we define another compact
group, we call the induced group of $(K,X)$, as

$\operatorname{Ind}(K,X):=K^{X}\rtimes{\rm Sym}(X),$

where the group ${\rm Sym}(X)$ of bijections of $X$ acts by permutations of
indices on $K^{X}:$

$\sigma((g_{x})_{x\in X})=(g_{\sigma^{-1}(x)})_{x\in X}\qquad\text{for
all}\quad\sigma\in{\rm Sym}(X),(g_{x})_{x\in X}\in K^{X}$

Observe that, if $\pi:K\to U(n)$ is a representation of $K$ on
$V={\mathbf{C}}^{n},$ then a unitary representation $\operatorname{Ind}(\pi)$
of $\operatorname{Ind}(K,X)$ on on $V^{X}$ is defined by

$\operatorname{Ind}(\pi)((g_{x})_{x\in X},\sigma)(v_{x})_{x\in
X}=(\pi(g_{x})v_{\sigma^{-1}(x)})_{x\in X},$

for $((g_{x})_{x\in X},\sigma)\in\operatorname{Ind}(K,X)$ and $(v_{x})_{x\in
X}\in V^{X}.$

Coming back to our setting, where $H$ is a closed subgroup of finite index in
$G$, we fix a transversal $X$ for the right cosets of $H;$ so, we have a
disjoint union $G=\bigsqcup_{x\in X}Hx$. For every $g\in G$ and $x\in X$, let
$x\cdot g$ and $c(x,g))$ be the unique elements in $X$ and $H$ such that
$xg\,=\,c(x,g)(x\cdot g).$ Observe that

$X\times G\to X,\qquad(x,g)\mapsto x\cdot g$

is an action of $G$ on $X$ (on the right), which is equivalent to the natural
action of $G$ on $H\backslash G$ given by right multiplication. In particular,
for every $g\in G,$ the map $\sigma(g):x\mapsto x\cdot g^{-1}$ belongs to
${\rm Sym}(X)$ and we have a homomorphism

$G\mapsto{\rm Sym}(X),\,\,g\mapsto\sigma(g).$

######  Proposition 9.

Let $(\operatorname{Bohr}(H),\beta)$ be the Bohr compactification of $H$. Let
$\operatorname{Ind}(\operatorname{Bohr}(H),X)$ be the compact group defined as
above. Consider the map
$\widetilde{\beta}:G\to\operatorname{Ind}(\operatorname{Bohr}(H),X)$ defined
by

$\widetilde{\beta}(g)=(\beta(c(x,g)))_{x\in X},\sigma(g))\qquad\text{for
all}\quad g\in G.$

The closure of $\widetilde{\beta}(G)$ in
$\operatorname{Ind}(\operatorname{Bohr}(H),X),$ together with the map
$\widetilde{\beta}$, is a Bohr compactification of $G.$

###### Proof.

It is readily checked that
$\widetilde{\beta}:G\to\operatorname{Ind}(\operatorname{Bohr}(H),X)$ is a
continuous homomorphism. Let $\rho:G\to U(n)$ be a finite dimensional unitary
representation of $G.$ Set $\pi:=\rho|_{H}\in\operatorname{Rep}_{\rm fd}(H).$
There exists $\pi^{\prime}\in\operatorname{Rep}_{\rm
fd}(\operatorname{Bohr}(H))$ such that $\pi=\pi^{\prime}\circ\beta$. Let
$\widetilde{\pi}:=\operatorname{Ind}_{H}^{G}\pi.$ As is well-known (see [BH,
1.F]), $\widetilde{\pi}$ can be realized on $V^{X}$ for $V:={\mathbf{C}}^{n}$
by the formula

$\widetilde{\pi}(g)(v_{x})_{x\in X})=(\pi(c(x,g))v_{x\cdot g})_{x\in
X}=(\pi(c(x,g))v_{\sigma(g^{-1})x})_{x\in X},$

for all $g\in G$ and $(v_{x})_{x\in X}\in V^{X}.$ With the unitary
representation $\operatorname{Ind}(\pi^{\prime})$ of
$\operatorname{Ind}(\operatorname{Bohr}(H),X)$ defined as above, we have
therefore

$None$

that is,
$\widetilde{\pi}=\operatorname{Ind}(\pi^{\prime})\circ\widetilde{\beta}.$ Now,

$\widetilde{\pi}=\operatorname{Ind}_{H}^{G}\pi=\operatorname{Ind}_{H}^{G}(\rho|_{H})$

is equivalent to the tensor product representation $\rho\otimes\lambda_{G/H}$,
where $\lambda_{G/H}$ is the regular representation of $G/H$ (see [BHV,
E.2.5]). Since $\lambda_{G/H}$ contains the trivial representation of $G,$ it
follows that $\rho$ is equivalent to a subrepresentation of $\widetilde{\pi}$;
so, we can identify $\rho$ with the representation of $G$ defined by a
$\widetilde{\pi}(G)$-invariant subspace $W$ of $V^{X}$. Denoting by $L$ the
closure of $\widetilde{\beta}(G)$, it follows from $(*)$ that $W$ is invariant
under $\operatorname{Ind}(\pi^{\prime})(L)$ and so defines a representation
$\rho^{\prime}$ of $L$. Then $\rho=\rho^{\prime}\circ\widetilde{\beta}$ and
the claim follows from Proposition 5. ∎

We will also need the following well-known (see [Hart-Kunen, Lemma 2.2])
description of the Bohr compactification of a quotient of $G$ in terms of the
Bohr compactification of $G$.

######  Proposition 10.

Let $(\operatorname{Bohr}(G),\beta)$ be the Bohr compactification of the
topological group $G$ and let $N$ be a closed normal subgroup of $G.$ Let
$K_{N}$ be the closure of $\beta(N)$ in $\operatorname{Bohr}(G)$

  * (i)

$K_{N}$ is a normal subgroup of $\operatorname{Bohr}(G)$ and $\beta$ induces a
continuous homomorphism $\overline{\alpha}:G/N\to\operatorname{Bohr}(G)/K_{N}$

  * (ii)

$(\operatorname{Bohr}(G)/K_{N},\overline{\alpha})$ is a Bohr compactification
of $G/N.$

###### Proof.

Let $(\operatorname{Bohr}(G/N),\overline{\beta})$ be the Bohr compactification
of $G/N.$ The canonical homomorphism $\alpha:G\to G/N$ induces a continuous
homomorphism
$\alpha^{\prime}:\operatorname{Bohr}(G)\to\operatorname{Bohr}(G/N)$ such that
the diagram

${G}$${\operatorname{Bohr}(G)}$${G/N}$${\operatorname{Bohr}(G/N)}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\overline{\beta}}$

commutes. It follows that $\beta(N)$ and hence $K_{N}$ is contained in ${\rm
Ker}\,\alpha^{\prime}.$ So, we have induced homomorphisms
$\overline{\alpha}:G/N\to\operatorname{Bohr}(G)/K_{N}$ and
$\overline{\alpha^{\prime}}:\operatorname{Bohr}(G)/K_{N}\to\operatorname{Bohr}(G/N),$
giving rise to a commutative diagram

${\operatorname{Bohr}(G)/K_{N}}$${G/N}$${\operatorname{Bohr}(G/N).}$$\scriptstyle{\overline{\alpha^{\prime}}}$$\scriptstyle{\overline{\alpha}}$$\scriptstyle{\overline{\beta}}$

It follows that $(\operatorname{Bohr}(G)/K_{N},\overline{\alpha})$ has the
same universal property for $G/N$ as
$(\operatorname{Bohr}(G/N),\overline{\beta})$. Since $\overline{\alpha}$ has
dense image, $(\operatorname{Bohr}(G)/K_{N},\overline{\alpha})$ is therefore a
Bohr compactification of $G/N.$ ∎

###  2.3. Bohr compactification of finitely generated abelian groups

Let $G$ be a locally compact abelian group. Its dual group ${\widehat{G}}$
consists of the continuous homomorphism from $G$ to the circle group
$\SS^{1};$ equipped with the topology of uniform convergence on compact
subsets, ${\widehat{G}}$ is again a locally compact abelian group. Let
${\widehat{G}}_{\rm disc}$ be the group $\widehat{G}$ equipped with the
discrete topology. It is well-known (see e.g. [BH, Proposition 4.C.4]) that
the Bohr compactification of $G$ coincides with the dual group $K$ of
${\widehat{G}}_{\rm disc}$, together with the embedding $i:G\to K$ given by
$i(g)(\chi)=\chi(g)$ for all $g\in G$ and $\chi\in{\widehat{G}}.$ Notice that
this implies that, by Pontrjagin duality, the dual group of
$\operatorname{Bohr}(G)$ coincides with ${\widehat{G}}_{\rm disc}$.

A more precise information on the structure of the Bohr compactification is
available in the case of a (discrete) finitely generated abelian group. As is
well-known, such a group $\Gamma$ splits a direct sum $\Gamma=F\oplus A$ of a
finite group $F$ (which is its torsion subgroup) and a free abelian group $A$
of finite rank $k\geq 0$, called the rank of $\Gamma$. Recall that
${\mathbf{Z}}_{p}$ denotes the ring of $p$-adic integers for a prime $p$ and
${\mathbf{A}}$ the ring of adèles over $\mathbf{Q}.$

######  Proposition 11.

Let $\Gamma$ be a finitely generated abelian group of rank $k.$

  * (i)

We have a direct sum decomposition

$\operatorname{Bohr}(\Gamma)\cong\operatorname{Bohr}(\Gamma)_{0}\oplus\operatorname{Prof}(\Gamma).$

  * (ii)

We have

$\operatorname{Prof}(\Gamma)\cong{F}\oplus\prod_{p\ \text{\rm
prime}}{\mathbf{Z}}_{p}^{k},$

where $F$ is a finite group.

  * (iii)

We have

$\operatorname{Bohr}(\Gamma)_{0}\cong\prod_{\omega\in\mathfrak{c}}{\mathbf{A}}^{k}/\mathbf{Q}^{k},$

a product of uncountably many copies of the adelic solenoid
${\mathbf{A}}^{k}/\mathbf{Q}^{k}$.

###### Proof.

We have $\Gamma\cong F\oplus{\mathbf{Z}}^{k}$ for a finite group $F$ and
$\operatorname{Bohr}({\mathbf{Z}}^{k})=\operatorname{Bohr}({\mathbf{Z}})^{k}.$
So, it suffices to determine $\operatorname{Bohr}({\mathbf{Z}}).$ As mentioned
above, $\operatorname{Bohr}({\mathbf{Z}})$ can be identified with the dual
group of the circle $\SS^{1}$ viewed as discrete group. Choose a linear basis
$\\{1\\}\cup\\{x_{\omega}\mid\omega\in\mathfrak{c}\\}$ of ${\mathbf{R}}$ over
$\mathbf{Q}$. Then $\SS^{1}\cong{\mathbf{R}}/{\mathbf{Z}}$ is isomorphic to
the abelian group

$(\mathbf{Q}/{\mathbf{Z}})\oplus\oplus_{\omega\in\mathfrak{c}}\mathbf{Q}.$

Hence,

$\operatorname{Bohr}({\mathbf{Z}})\cong\widehat{\mathbf{Q}/{\mathbf{Z}}}\oplus\prod_{\omega\in\mathfrak{c}}\widehat{\mathbf{Q}}.$

Now,

$\mathbf{Q}/{\mathbf{Z}}=\oplus_{p\ \text{prime}}Z(p^{\infty}),$

with $Z(p^{\infty})=\varinjlim_{k}{\mathbf{Z}}/p^{k}{\mathbf{Z}}$ the
$p$-primary component of $\mathbf{Q}/{\mathbf{Z}}.$ Hence,

$\widehat{Z(p^{\infty})}\cong\varprojlim_{k}{\mathbf{Z}}/p^{k}{\mathbf{Z}}={\mathbf{Z}}_{p}.$

On the other hand, $\widehat{\mathbf{Q}}$ can be identified with the solenoid
${\mathbf{A}}/\mathbf{Q}$ (see e.g. [HewittRoss, (25.4)]). It follows that

$\operatorname{Bohr}(\Gamma)\cong\prod_{p\ \text{\rm
prime}}{\mathbf{Z}}_{p}\oplus\prod_{\omega\in\mathfrak{c}}{\mathbf{A}}/\mathbf{Q}.$

∎

###  2.4. Restrictions of representations to normal subgroups

Let $\Gamma$ be a group and $N$ a normal subgroup of $\Gamma$. Recall that
$\Gamma$ acts on $\widehat{N}_{\rm fd}$: for $\sigma\in\widehat{N}_{\rm fd}$
and $\gamma\in\Gamma,$ the conjugate representation
$\sigma^{\gamma}\in\widehat{N}_{\rm fd}$ is defined by

$\sigma^{\gamma}(n)\,=\,\sigma(\gamma^{-1}n\gamma),\hskip 14.22636pt\text{for
all}\hskip 5.69046ptn\in N.$

The stabilizer $\Gamma_{\sigma}$ of $\sigma$ is the subgroup consisting of all
$\gamma\in\Gamma$ for which $\sigma^{\gamma}$ is equivalent $\sigma;$ observe
that $\Gamma_{\sigma}$ contains $N.$

Given a unitary representation $\rho$ of $N$ on a finite dimensional vector
space $V$ and $\sigma\in\widehat{N}_{\rm fd},$ we denote by $V^{\sigma}$ the
_$\sigma$ -isotypical component_ of $\rho,$ that is, the sum of all
$\rho$-invariant subspaces $W$ for which the restriction of $\rho$ to $W$ is
equivalent to $\sigma.$ Observe that $V$ decomposes as direct sum
$V=\oplus_{\sigma\in\Sigma_{\rho}}V^{\sigma}$, where $\Sigma_{\rho}$ is the
finite set of $\sigma\in\widehat{N}_{\rm fd}$ with $V^{\sigma}\neq\\{0\\}.$

######  Proposition 12.

Let $\pi$ be an irreducible unitary representation of $\Gamma$ on a finite
dimensional vector space $V.$ Let
$V=\oplus_{\sigma\in\Sigma_{\pi|_{N}}}V^{\sigma}$ be the decomposition of the
restriction $\pi|_{N}$ of $\pi$ to $N$ into isotypical components. Then
$\Sigma_{\pi|_{N}}$ coincides with a $\Gamma$-orbit: there exists
$\sigma\in\widehat{N}_{\rm fd}$ such that
$\Sigma_{\pi|_{N}}=\\{\sigma^{\gamma}\ :\ \gamma\in\Gamma\\};$ in particular,
$\Gamma_{\sigma}$ has finite index in $\Gamma.$

###### Proof.

Let $\sigma\in\Sigma_{\pi|_{N}}$ and fix a transversal $T$ for the left cosets
of $\Gamma_{\sigma}$ with $e\in T$. Then $V^{\sigma^{t}}=\pi(t)V^{\sigma}$ for
all $t\in T$ Since $\pi$ is irreducible and $\sum_{t\in T}\pi(t)V^{\sigma}$ is
$\pi(\Gamma)$-invariant, it follows that $\Sigma_{\pi|_{N}}$ is a
$\Gamma$-orbit. ∎

##  3\. Proof of Theorem 1

###  3.1. Distortion and Bohr compactification

Let $\Gamma$ be a finitely generated group with a finite set $S$ of
generators. For $\gamma\in\Gamma$, denote by $\ell_{S}(\gamma)$ the word
length of $\gamma$ with respect to $S\cup S^{-1}$ and set

$t(\gamma)=\liminf_{n\to\infty}\dfrac{\ell_{S}(\gamma^{n})}{n}.$

The number $t(\gamma)$ is called the _translation number_ of $\gamma$ in
[Gersten]

######  Definition 13.

An element $\gamma\in\Gamma$ is said to be distorted if $t(\gamma)=0.$

In fact, since the sequence $n\mapsto\ell_{S}(\gamma^{n})$ is subadditive, we
have, by Fekete’s lemma,

$t(\gamma)=\lim_{n\to\infty}\dfrac{\ell_{S}(\gamma^{n})}{n}=\inf\left\\{\dfrac{\ell_{S}(\gamma^{n})}{n}:n\in{\mathbf{N}}^{*}\right\\}$

The property of being distorted is independent of the choice of the set of
generators. Distorted elements are called _algebraically parabolic_ in
[Gromov, (7.5), p.90], but we prefer to use the terminology from [Franks]. The
relevance of distorsion to the Bohr compactification lies in the following
proposition; for a related result with a similar proof, see [LMR, (2.4)].

######  Proposition 14.

Let $\Gamma$ be a finitely generated group and $\gamma\in\Gamma$ a distorted
element. Then, for every finite dimensional unitary representation
$\pi:\Gamma\to U(N)$ of $\Gamma,$ the matrix $\pi(\gamma)\in U(N)$ has finite
order.

###### Proof.

It suffices to show that all eigenvalues of the unitary matrix $\pi(\gamma)$
are roots of unity. Assume, by contradiction, that $\pi$ has an eigenvalue
$\lambda\in\SS^{1}$ of infinite order.

Let $S$ be a finite set of generators of $\Gamma$ with $S=S^{-1}.$ The group
$\pi(\Gamma)$ is generated by the set $\\{\pi(s)\mid s\in S\\}.$ Hence,
$\pi(G)$ is contained in $GL_{N}(L)$, where $L$ is the subfield of
${\mathbf{C}}$ generated by the matrix coefficients of the $\pi(s)$’s. It
follows that $\lambda$ is contained in a finitely generated extension $\ell$
of $L$. By a lemma of Tits ([Tits, Lemma 4.1]), there exists a locally compact
field $k$ endowed with an absolute value $|\cdot|$ and a field embedding
$\sigma:\ell\to k$ such that $|\sigma(\lambda)|\neq 1.$ Upon replacing
$\gamma$ by $\gamma^{-1}$, we may assume that $|\sigma(\lambda)|>1.$

Define a function (“norm”) $\xi\mapsto\|\xi\|$ on $k^{N}$ by

$\|\xi\|=\max\\{|\xi_{1}|,\dots,|\xi_{N}|\\}\qquad\text{for
all}\quad\xi=(\xi_{1},\dots,\xi_{N})\in k^{N}.$

For a matrix $A\in GL_{N}(k),$ set $\|A\|=\sup_{\xi\neq 0}\|A\xi\|/\|\xi\|.$
It is obvious that $\|A\xi\|\leq\|A\|\|\xi\|$ for all $\xi\in k^{N}$ and hence

$None$ $\|AB\|\leq\|A\|\|B\|\qquad\text{for all}\quad A,B\in GL_{N}(k).$

In particular, we have $\|A^{n}\|\leq\|A\|^{n}$ for all $A\in GL_{N}(k)$ and
$n\in{\mathbf{N}}.$

For a matrix $w\in GL_{n}(\ell),$ denote by $\sigma(w)$ the matrix in
$GL_{n}(k)$ obtained by applying $\sigma$ to the entries of $w.$ Set
$A_{s}=\sigma(\pi(s))$ for $s\in S$ and $A:=\sigma(\pi(\gamma)).$ With

$C:=\max\\{\|A_{s}\|:s\in S\\},$

it is clear that Inequality $(*)$ implies that

$None$ $\|A^{n}\|=\|\sigma(\pi(\gamma^{n}))\|\leq
C^{\ell_{S}(\gamma^{n})}\qquad\text{for all}\quad n\in{\mathbf{N}}.$

On the other hand, $\sigma(\lambda)$ is an eigenvalue of $A$; so, there exists
$\xi\in k^{N}\setminus\\{0\\}$ such that $A\xi=\sigma(\lambda)\xi$ and hence
$A^{n}\xi=\sigma(\lambda)^{n}\xi$ for all $n\in{\mathbf{N}}.$ So, for every
$n\in{\mathbf{N}},$ we have

$\|A^{n}\xi\|=|\sigma(\lambda)|^{n}\|\xi\|$

and this implies that

$\|A^{n}\|\geq|\sigma(\lambda)|^{n}.$

In view of $(**)$, we obtain therefore

$\dfrac{\ell_{S}(\gamma^{n})\log
C}{n}\geq{\log|\sigma(\lambda)|}\qquad\text{for all}\quad n\in{\mathbf{N}}.$

Since $|\sigma(\lambda)|>1,$ this contradicts the fact that
$\liminf_{n\to\infty}\dfrac{\ell_{S}(\gamma^{n})}{n}=0.$ ∎

###  3.2. Distorted elements in nilpotent groups

Let $\Gamma$ be a finitely generated nilpotent subgroup. For subsets $A,B$ in
$\Gamma,$ we let $[A,B]$ denote the subgroup of $\Gamma$ generated by all
commutators $[a,b]=aba^{-1}b^{-1}$, for $a\in A$ and $b\in B.$ Let

$\Gamma^{(0)}\supset\Gamma^{(1)}\supset\cdots\supset\Gamma^{(d-1)}\supset\Gamma^{(d)}=\\{e\\}$

be the lower central series of $\Gamma$, defined inductively by
$\Gamma^{(0)}=\Gamma$ and $\Gamma^{(k+1)}=[\Gamma^{(k)},\Gamma].$ The step of
nilpotency of $\Gamma$ is the smallest $d\geq 1$ such that
$\Gamma^{(d-1)}\neq\\{e\\}$ and $\Gamma^{(d)}=\\{e\\}$.

######  Proposition 15.

Let $\Gamma$ be a finitely generated nilpotent subgroup. Every
$\gamma\in\Gamma^{(1)}=[\Gamma,\Gamma]$ is distorted.

###### Proof.

Let $S$ be a finite set of generators of $\Gamma$ with $S=S^{-1}.$ Let $d\geq
1$ be the step of nilpotency of $\Gamma$. The case $d=1$ being trivial, we
will assume that $d\geq 2.$ We will show by induction on
$i\in\\{1,\dots,d-1\\}$ that every $\gamma\in\Gamma^{(d-i)}$ is distorded.

$\bullet$ First step. Assume that $i=1$. It is well-known that every element
$\gamma$ in $\Gamma^{(d-1)}$ is distorted (see for instance [Gromov, (7.6), p.
91]); in fact, more precise estimates are available: for every
$\gamma\in\Gamma^{(d-1)},$ we have $\ell_{S}(\gamma^{n})=O(n^{1/d})$ as
$n\to\infty$ (see [Tits2, 2.3 Lemme] or [Drutu, Lemma 14.15]).

$\bullet$ Second step. Assume that, for every finitely generated nilpotent
subgroup $\Lambda$ of step $d^{\prime}\geq 2$, every element
$\delta\in\Lambda^{(d^{\prime}-i)}$ is distorded for
$i\in\\{1,\dots,d^{\prime}-2\\}.$ Let $\gamma\in\Gamma^{(d-i-1)}$ and fix
$\varepsilon>0.$

The quotient group $\overline{\Gamma}=\Gamma/\Gamma^{(d-1)}$ is nilpotent of
step $d^{\prime}=d-1$ and $p(\gamma)\in\overline{\Gamma}^{(d^{\prime}-i)}$,
where $p:\Gamma\to\overline{\Gamma}$ is the quotient map. By induction
hypothesis, $p(\gamma)$ is distorted in $\overline{\Gamma}$ with respect to
the generating set $\overline{S}:=p(S).$ So, we have
$\lim_{n\to\infty}\dfrac{\ell_{\overline{S}}(p(\gamma)^{n})}{n}=0;$ hence, we
can find an integer $N\geq 1$ such that

$None$ $\forall n\geq N,\exists\delta_{n}\in\Gamma^{(d-1)}\ :\
\dfrac{\ell_{S}(\gamma^{n}\delta_{n})}{n}\leq\varepsilon.$

By the first step, we have
$\lim_{k\to\infty}\dfrac{\ell_{S}(\delta_{N}^{k})}{k}=0,$ since
$\delta_{N}\in\Gamma^{(d-1)};$ so, there exists $K\geq 1$ such that

$None$ $\forall k\geq K\ :\
\dfrac{\ell_{S}(\delta_{N}^{k})}{k}\leq\varepsilon.$

Let $k\geq K.$ We have

$None$
$\dfrac{\ell_{S}(\gamma^{Nk})}{Nk}=\dfrac{\ell_{S}((\gamma^{Nk}\delta_{N}^{k})(\delta_{N}^{-1})^{k})}{Nk}\leq\dfrac{\ell_{S}(\gamma^{Nk}\delta_{N}^{k})}{Nk}+\dfrac{\ell_{S}(\delta_{N}^{k})}{Nk}.$

Now, since $\Gamma^{(d-1)}$ is contained in the center of $\Gamma,$ the
elements $\delta_{N}$ and $\gamma_{N}$ commute and hence, by $(*)$, we have

$\dfrac{\ell_{S}(\gamma^{Nk}\delta_{N}^{k})}{Nk}=\dfrac{\ell_{S}((\gamma^{N}\delta_{N})^{k})}{Nk}\leq
k\dfrac{\ell_{S}(\gamma^{N}\delta_{N})}{Nk}=\dfrac{\ell_{S}(\gamma^{N}\delta_{N})}{N}\leq\varepsilon.$

So, together with $(***)$ and $(**)$, we obtain

$\forall k\geq K\ :\ \dfrac{\ell_{S}(\gamma^{Nk})}{Nk}\leq 2\varepsilon.$

This shows that $t(\gamma)=0.$ ∎

###  3.3. Congruence subgroups in unipotent groups

The following result, which shows that the congruence subgroup problem has a
positive solution for unipotent groups, is well-known (see the sketch in
[Raghunathan-CSP, p.108]); for the convenience of the reader, we reproduce its
short proof.

######  Proposition 16.

Let $\mathbf{U}$ be a unipotent algebraic group over $\mathbf{Q}.$ Let
$\Gamma$ be an arithmetic subgroup of $\mathbf{U}(\mathbf{Q}).$ Then every
finite index subgroup of $\Gamma$ is a congruence subgroup.

###### Proof.

We can find a sequence

$\mathbf{U}=\mathbf{U}_{0}\supset\mathbf{U}_{1}\supset\cdots\supset\mathbf{U}_{d-1}\supset\mathbf{U}_{d}=\\{e\\}$

of normal $\mathbf{Q}$-subgroups of $\mathbf{U}$ such that the factor groups
$\mathbf{U}_{i}/\mathbf{U}_{i+1}$ are $\mathbf{Q}$-isomorphic to
$\mathbf{G}_{a}$, the additive group of dimension 1 (see [Borel, (15.5)]).

We proceed by induction on $d\geq 1$. If $d=1,$ then $\Gamma$ is commensurable
to ${\mathbf{Z}}$ and the claim is obvious true. Assume that $d\geq 2.$ Then
$\mathbf{U}$ can be written as semi-direct product
$\mathbf{U}=\mathbf{U}_{1}\rtimes\mathbf{G}_{a}.$ By [Borel-HC, Corollary
4.6], $\Gamma$ is commensurable to
$\mathbf{U}_{1}({\mathbf{Z}})\rtimes{\mathbf{Z}}.$ Let $H$ a subgroup of
finite index in $\Gamma.$ Then $H\cap\mathbf{U}_{1}({\mathbf{Z}})$ has finite
index in $\mathbf{U}_{1}({\mathbf{Z}})$ and hence, by induction hypothesis,
contains the kernel of the reduction
$\mathbf{U}_{1}({\mathbf{Z}})\to\mathbf{U}_{1}({\mathbf{Z}}/N_{1}{\mathbf{Z}})$
modulo some $N_{1}\geq 1.$ Moreover, $H\cap{\mathbf{Z}}=N_{2}{\mathbf{Z}}$ for
some $N_{2}\geq 1.$ Hence, $H$ contains the kernel of the reduction
$\mathbf{U}({\mathbf{Z}})\to\mathbf{U}({\mathbf{Z}}/N_{1}N_{2}{\mathbf{Z}})$
modulo $N_{1}N_{2}.$ ∎

###  3.4. Proof of Theorem 1

Let $\Gamma$ be a finitely generated nilpotent group and
$\alpha:\Gamma\to\operatorname{Prof}(\Gamma)$ the canonical homomorphism.
Recall (see Proposition 11) that the Bohr compactification of $\Gamma^{\rm
Ab}=\Gamma/[\Gamma,\Gamma]$ splits as a direct sum

$\operatorname{Bohr}(\Gamma^{\rm Ab})=\operatorname{Bohr}(\Gamma^{\rm
Ab})_{0}\oplus B_{1},$

for a closed subgroup $B_{1}\cong\operatorname{Prof}(\Gamma^{\rm Ab}).$ Let
$p:\operatorname{Bohr}(\Gamma^{\rm Ab})\to\operatorname{Bohr}(\Gamma^{\rm
Ab})_{0}$ be the corresponding projection. Denote by
$\beta_{0}:\Gamma\to\operatorname{Bohr}(\Gamma^{\rm Ab})$ the map induced by
the quotient homomorphism $\Gamma\to\Gamma^{\rm Ab}$. Set

$K:=\operatorname{Bohr}(\Gamma^{\rm
Ab})_{0}\times\operatorname{Prof}(\Gamma),$

and let $\beta:\Gamma\to K$ be the homomorphism
$\gamma\mapsto(p\circ\beta_{0}(\gamma),\alpha(\gamma)).$ We claim that
$(K,\beta)$ is a Bohr compactification for $\Gamma.$

$\bullet$ First step. We claim that $\beta(\Gamma)$ is dense in $K.$ Indeed,
let $L$ be the closure of $\beta(\Gamma)$ in $K$ and $L_{0}$ its connected
component. Since $\operatorname{Prof}(\Gamma)$ is totally disconnected, the
projection of $L_{0}$ on $\operatorname{Prof}(\Gamma)$ is trivial; hence
$L_{0}=K_{0}\times\\{1\\}$ for a connected closed subgroup $K_{0}$ of
$\operatorname{Bohr}(\Gamma^{\rm Ab})_{0}.$ The projection of $L$ on
$\operatorname{Bohr}(\Gamma^{\rm Ab})_{0}$ induces then a continuous
homomorphism

$f:L/L_{0}\to\operatorname{Bohr}(\Gamma^{\rm Ab})_{0}/K_{0}.$

Observe that $f$ has dense image, since $p\circ\beta_{0}(\Gamma)$ is dense in
$\operatorname{Bohr}(\Gamma^{\rm Ab})_{0}$; so, $f$ is surjective by
compactness of $L/L_{0}.$ It follows, by compactness again, that
$\operatorname{Bohr}(\Gamma^{\rm Ab})_{0}/K_{0}$ is topologically isomorphic
to a quotient of $L/L_{0}.$ As $L/L_{0}$ is totally disconnected, this implies
(see [Bourbaki, Chap. 3, §4, Corollaire 3]) that
$\operatorname{Bohr}(\Gamma^{\rm Ab})_{0}/K_{0}$ is also totally disconnected
and hence that $K_{0}=\operatorname{Bohr}(\Gamma^{\rm Ab})_{0}.$ So,
$\operatorname{Bohr}(\Gamma^{\rm Ab})_{0}\times\\{1\\}$ is contained in $L.$
It follows that $L$ is the product of $\operatorname{Bohr}(\Gamma^{\rm
Ab})_{0}$ with a subgroup of $\operatorname{Prof}(\Gamma).$ Since
$\alpha(\Gamma)$ is dense in $\operatorname{Prof}(\Gamma),$ this subgroup
coincides with $\operatorname{Prof}(\Gamma)$, that is, $L=K$ and the claim is
proved.

$\bullet$ Third step. We claim that every irreducible unitary representation
$\pi:\Gamma\to U(N)$ of $\Gamma$ is of the form $\chi\otimes\rho$ for some
$\chi\in\widehat{\Gamma^{\rm Ab}}$ and $\rho\in\widehat{\Gamma}_{\rm finite}.$

Indeed, Propositions 14 and 15, imply that $\pi([\Gamma,\Gamma])$ is a
periodic subgroup of $U(N).$ Since $\Gamma$ is finitely generated,
$[\Gamma,\Gamma]$ is finitely generated (in fact, every subgroup of $\Gamma$
is finitely generated; see [Raghunathan, 2.7 Theorem]). Hence, by Schur’s
theorem (see [Wehrfritz, 4.9 Corollary]), $\pi([\Gamma,\Gamma])$ is finite. It
follows that there exists a finite index normal subgroup $H$ of
$[\Gamma,\Gamma]$ so that $\pi|_{H}$ is the trivial representation of $H$.

Next, we claim that there exists a normal subgroup $\Delta$ of finite index in
$\Gamma$ such that $\Delta\cap[\Gamma,\Gamma]=H.$ Indeed, since
$\Gamma/[\Gamma,\Gamma]$ is abelian and finitely generated, we have
$\Gamma/[\Gamma,\Gamma]\cong{\mathbf{Z}}^{k}\oplus F$ for some finite subgroup
$F$ and some integer $k\geq 0.$ Let $\Gamma_{1}$ be the inverse image in
$\Gamma$ of the copy of ${\mathbf{Z}}^{k}$ in $\Gamma/[\Gamma,\Gamma].$ Then
$\Gamma_{1}$ is a normal subgroup of finite index of $\Gamma.$ Moreover,
$\Gamma_{1}$ can be written as iterated semi-direct product

$\Gamma_{1}=(\dots(([\Gamma,\Gamma]\rtimes{\mathbf{Z}})\rtimes{\mathbf{Z}})\rtimes{\mathbf{Z}})).$

Set

$\Delta:=(\dots((H\rtimes{\mathbf{Z}})\rtimes{\mathbf{Z}})\rtimes{\mathbf{Z}})).$

Then $\Delta$ is a normal subgroup of finite index of $\Gamma$ with
$\Delta\cap[\Gamma,\Gamma]=H.$

Since $\pi|_{H}$ is trivial on $H$ and since $[\Delta,\Delta]\subset H,$ the
restriction $\pi|_{\Delta}$ of $\pi$ to $\Delta$ factorizes through
$\Delta^{\rm Ab}.$ So, by Proposition 12, there exists a finite $\Gamma$-orbit
$\mathcal{O}$ in $\widehat{\Delta^{\rm Ab}}$ such that we have a direct sum
decomposition $V=\bigoplus_{\chi\in\mathcal{O}}V^{\chi}$, where $V^{\chi}$ is
the $\chi$-isotypical component of $\pi|_{\Delta}.$

Fix $\chi\in\mathcal{O}.$ Since $\chi$ is trivial on $H$ and since
$\Delta\cap[\Gamma,\Gamma]=H,$ we can view $\chi$ as a unitary character of
the subgroup $\Delta/(\Delta\cap[\Gamma,\Gamma])$ of $\Gamma^{\rm Ab}.$ Hence,
$\chi$ extends to a character $\widetilde{\chi}\in\widehat{\Gamma^{\rm Ab}}$
(see, e.g. [HewittRoss, (24.12)]). This implies that $\Gamma_{\chi}=\Gamma$;
indeed,

$\chi^{\gamma}(\delta)=\widetilde{\chi}(\gamma^{-1}\delta\gamma)=\widetilde{\chi}(\delta)=\chi(\delta)$

for every $\gamma\in\Gamma$ and $\delta\in\Delta.$ This shows that
$\mathcal{O}$ is a singleton and so $V=V^{\chi}.$ We write

$\pi=\widetilde{\chi}\otimes(\overline{\widetilde{\chi}}\otimes\pi).$

Then $\rho:=\overline{\widetilde{\chi}}\otimes\pi$ is an irreducible unitary
representation of $\Gamma$ which is trivial on $\Delta$; so, $\rho$ has finite
image and $\pi=\widetilde{\chi}\otimes\rho.$

$\bullet$ Third step. Let $\pi\in\widehat{\Gamma}_{\rm fd}.$ We claim that
there exists a representation $\pi^{\prime}\in\widehat{K}$ such that
$\pi=\pi^{\prime}\circ\beta.$ Once proved, Proposition 5 will imply that
$(K,\beta)$ is a Bohr compactification for $\Gamma$.

By the second step, we can write $\pi=\chi\otimes\rho$ for some
$\chi\in\widehat{\Gamma^{\rm Ab}}$ and $\rho\in\widehat{\Gamma}_{\rm finite}.$
On the one hand, we can write $\rho=\rho^{\prime}\circ\alpha$ for some
$\rho^{\prime}\in\widehat{\operatorname{Prof}(\Gamma)}$, by the universal
property of $\operatorname{Prof}(\Gamma).$ On the other hand, we can decompose
$\chi$ as $\chi=\chi_{0}\chi_{1}$ with $\chi_{0}\in\widehat{\Gamma^{\rm Ab}}$
of infinite order and $\chi_{1}\in\widehat{\Gamma^{\rm Ab}}$ of finite order.
We have $\chi_{0}=\chi^{\prime}_{0}\circ(p\circ\beta_{0})$ and
$\chi_{1}=\chi_{1}^{\prime}\circ\alpha$ for unitary characters
$\chi^{\prime}_{0}$ of $\operatorname{Bohr}(\Gamma^{\rm Ab})_{0}$ and
$\chi_{1}^{\prime}$ of $\operatorname{Prof}(\Gamma^{\rm Ab})$. For
$\pi^{\prime}=\chi_{0}\otimes(\chi^{\prime}_{1}\otimes\rho^{\prime})$, we have
$\pi^{\prime}\in\widehat{K}$ and $\pi=\pi^{\prime}\circ\beta.$

##  4\. Proof of Theorems 2

Let $\mathbf{G}=\mathbf{U}\rtimes\mathbf{H}$ be a Levi decomposition of
$\mathbf{G}$ and set

$\Lambda=\mathbf{H}({\mathbf{Z}}),\quad\Delta=\mathbf{U}({\mathbf{Z}}),\qquad\text{and}\qquad\Gamma=\Delta\rtimes\Lambda.$

Denote by $\beta_{\Delta}:\Delta\to\operatorname{Bohr}(\Delta)$ and
$\beta_{\Lambda}:\Lambda\to\operatorname{Bohr}(\Lambda)$ the natural
homomorphisms. Observe that, by the universal property of
$\operatorname{Bohr}(\Delta),$ every element $\lambda\in\Lambda$ defines a
continuous automorphism $\theta_{b}(\lambda)$ of $\operatorname{Bohr}(\Delta)$
such that

$\theta_{b}(\lambda)(\delta)=\beta_{\Delta}(\lambda\delta\lambda^{-1})\qquad\text{for
all}\quad\delta\in\Delta.$

The corresponding homomorphism $\theta_{b}:\Lambda\to{\rm
Aut}(\operatorname{Bohr}(\Delta))$ defines an action of $\Lambda$ on
$\operatorname{Bohr}(\Delta).$ By Theorem 1, we have

$\operatorname{Bohr}(\Delta)=\operatorname{Bohr}(\Delta^{\rm
Ab})_{0}\times\operatorname{Prof}(\Delta).$

The group $\Lambda$ acts naturally on $\Delta^{\rm Ab}$ and, by duality, on
$\widehat{\Delta^{\rm Ab}}.$ Let

$H:=\widehat{\Delta^{\rm Ab}}_{\Lambda-\rm{fin}}\subset\widehat{\Delta^{\rm
Ab}}$

be the subgroup of characters of $\Delta^{\rm Ab}$ with finite
$\Lambda$-orbits. Observe that $H$ contains the torsion subgroup of
$\widehat{\Delta^{\rm Ab}}.$

Let

$\alpha:\Lambda\to{\rm Aut}(H)$

be the homomorphism given by the action of $\Lambda$ on $H.$

For a locally compact group $G,$ the group ${\rm Aut}(G)$ of continuous
automorphisms of $G$ will be endowed with the compact-open topology for which
it is also a (not necessarily locally compact) topological group (see
[HewittRoss, (26.3)]).

$\bullet$ First step. We claim that the closure of $\alpha(\Lambda)$ in ${\rm
Aut}(H)$ is compact. Indeed, let us identify ${\rm Aut}(H)$ with a subset of
the product space $H^{H}$. The topology of ${\rm Aut}(H)$ coincides with the
topology induced by the product topology on $H^{H}.$ Viewed this way,
$\alpha(\Lambda)$ is a subspace of the product $\prod_{\chi\in
H}\chi^{\Lambda}$ of the finite $\Lambda$-orbits $\chi^{\Lambda}.$ Since
$\prod_{\chi\in H}\chi^{\Lambda}$ is compact and hence closed, the claim is
proved.

Next, let $N$ be the annihilator of $H$ in $\operatorname{Bohr}(\Delta^{\rm
Ab}).$ Then $N$ is $\Lambda$-invariant and the induced action of $\Lambda$ on
$\operatorname{Bohr}(\Delta^{\rm Ab})/N$ is a quotient of the action given by
$\theta_{b}.$

Let $C$ be the connected component of $\operatorname{Bohr}(\Delta^{\rm
Ab})/N.$ Then $C$ coincides with the image of $\operatorname{Bohr}(\Delta^{\rm
Ab})_{0}$ in $\operatorname{Bohr}(\Delta^{\rm Ab})/N$ (see [Bourbaki, Chap. 3,
§4, Corollaire 3]) and so

$C\cong\operatorname{Bohr}(\Delta^{\rm
Ab})_{0}/(N\cap\operatorname{Bohr}(\Delta^{\rm Ab})_{0}).$

Since $C$ is invariant under $\Lambda$, we obtain an action of $\Lambda$ on
$C;$ let

$\widehat{\alpha}:\Lambda\to{\rm Aut}(C)$

be the corresponding homomorphism.

$\bullet$ Second step. We claim that the action $\widehat{\alpha}$ of
$\Lambda$ on $C$ extends to an action of $\operatorname{Bohr}(\Lambda)$; more
precisely, there exists a continuous homomorphism

$\widehat{\alpha}^{\prime}:\operatorname{Bohr}(\Lambda)\to{\rm Aut}(C)$

such that the diagram

${\operatorname{Bohr}(\Lambda)}$${\Lambda}$${{\rm
Aut}(C)}$$\scriptstyle{\widehat{\alpha}^{\prime}}$$\scriptstyle{\beta_{\Lambda}}$$\scriptstyle{\widehat{\alpha}}$

commutes. Indeed, by the first step, the closure $K$ of $\alpha(\Lambda)$ in
${\rm Aut}(H)$ is a compact group. Hence, by the universal property of
$\operatorname{Bohr}(\Lambda),$ there exists a continuous homomorphism

$\alpha^{\prime}:\operatorname{Bohr}(\Lambda)\to K\subset{\rm Aut}(H)$

such that the diagram

${\operatorname{Bohr}(\Lambda)}$${\Lambda}$${{\rm
Aut}(H)}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta_{\Lambda}}$$\scriptstyle{\alpha}$

commutes. Since $\widehat{H}=\operatorname{Bohr}(\Delta^{\rm Ab})/N,$ we
obtain by duality a continuous homomorphism
$\widehat{\alpha}^{\prime}:\operatorname{Bohr}(\Lambda)\to{\rm
Aut}(\operatorname{Bohr}(\Delta^{\rm Ab})/N)$. The connected component $C$ of
$\operatorname{Bohr}(\Delta^{\rm Ab})/N$ is invariant under
$\operatorname{Bohr}(\Lambda)$. This proves the existence of the map
$\widehat{\alpha}^{\prime}:\operatorname{Bohr}(\Lambda)\to{\rm Aut}(C)$ with
the claimed property.

Next, observe that, by the universal property of
$\operatorname{Prof}(\Delta),$ every element $\lambda\in\Lambda$ defines a
continuous automorphism $\theta_{p}(\lambda)$ of $\operatorname{Prof}(\Delta)$
such that

$\theta_{p}(\lambda)(\delta)=\beta_{\Delta}(\lambda\delta\lambda^{-1})\qquad\text{for
all}\quad\delta\in\Delta.$

The corresponding homomorphism $\theta_{p}:\Lambda\to{\rm
Aut}(\operatorname{Prof}(\Delta))$ defines an action of $\Lambda$ on
$\operatorname{Prof}(\Delta).$

$\bullet$ Third step. We claim that the action $\theta_{p}$ of $\Lambda$ on
$\operatorname{Prof}(\Delta)$ extends to an action of
$\operatorname{Bohr}(\Lambda);$ more precisely, there exists a homomorphism
$\theta^{\prime}:\operatorname{Bohr}(\Lambda)\to{\rm
Aut}(\operatorname{Prof}(\Delta))$ such that the diagram

${\operatorname{Prof}(\Lambda)}$${\Lambda}$${{\rm
Aut}(\operatorname{Prof}(\Delta))}$$\scriptstyle{\theta^{\prime}}$$\scriptstyle{\beta_{\Lambda}}$$\scriptstyle{\theta_{p}}$

commutes. Indeed, since $\Delta$ is finitely generated and since its image in
$\operatorname{Bohr}(\Delta)$ dense, the profinite group
$\operatorname{Bohr}(\Delta)$ is finitely generated (that is, there exists a
finite subset of $\operatorname{Bohr}(\Delta)$ which generates a dense
subgroup). This implies that ${\rm Aut}(\operatorname{Bohr}(\Delta))$ is a
profinite group (see [Ribes-Zal, Corollary 4.4.4]) and so there exists a
homomorphism $\theta^{\prime}_{p}:\operatorname{Prof}(\Lambda)\to{\rm
Aut}(\operatorname{Prof}(\Delta))$ such that
$\theta^{\prime}_{p}\circ\alpha_{\Lambda}=\theta_{p}.$ We then lift
$\theta^{\prime}_{p}$ to a homomorphism
$\theta^{\prime}:\operatorname{Bohr}(\Lambda)\to{\rm
Aut}(\operatorname{Prof}(\Delta)).$

We set

$Q:=\operatorname{Bohr}(\Delta)/(N\cap\operatorname{Bohr}(\Delta^{\rm
Ab})_{0})=C\times\operatorname{Prof}(\Delta);$

we have an action of $\Lambda$ on $Q$ given by the homomorphism

$\widehat{\alpha}\oplus\theta_{p}:\Lambda\to{\rm Aut}(C)\times{\rm
Aut}(\operatorname{Prof}(\Delta))\subset{\rm Aut}(Q)$

and, by the second and third step, an action of $\operatorname{Bohr}(\Lambda)$
on $Q$ given by

$\widehat{\alpha}^{\prime}\oplus\theta^{\prime}:\operatorname{Bohr}(\Lambda)\to{\rm
Aut}(C)\times{\rm Aut}(\operatorname{Prof}(\Delta))$

such that the diagram

${\operatorname{Bohr}(\Lambda)}$${\Lambda}$${{\rm Aut}(C)\times{\rm
Aut}(\operatorname{Prof}(\Delta))}$$\scriptstyle{\widehat{\alpha}^{\prime}\oplus\theta^{\prime}}$$\scriptstyle{\beta_{\Lambda}}$$\scriptstyle{\widehat{\alpha}\oplus\theta_{p}}$

commutes.

Let

$B:=(C\times\operatorname{Prof}(\Delta))\rtimes\operatorname{Bohr}(\Lambda)$

be the semi-direct product defined by
$\widehat{\alpha}^{\prime}\oplus\theta^{\prime}.$ Let

$p:\operatorname{Bohr}(\Delta)\to C=\operatorname{Bohr}(\Delta^{\rm
Ab})_{0}/(N\cap\operatorname{Bohr}(\Delta^{\rm Ab})_{0})$

be the quotient epimorphism.

$\bullet$ Fourth step. We claim that $B,$ together with the map
$\beta:\Gamma\to B,$ given by

$\beta(\delta,\lambda)=(p(\beta_{\Delta}(\delta)),\beta_{\Lambda}(\lambda))\qquad\text{for
all}\quad(\delta,\lambda)\in\Gamma,$

is a Bohr compactification for $\Gamma=\Delta\rtimes\Lambda.$

First, we have to check that $\beta$ is a homomorphism with dense image. Since
$p\circ\beta_{\Delta}$ and $\beta_{\Lambda}$ are homomorphisms with dense
image, it suffices to show that

$\beta(\lambda\delta\lambda^{-1},e)=((\widehat{\alpha}^{\prime}\oplus\theta^{\prime})(\beta_{\Lambda}(\lambda))(p(\beta_{\Delta}(\delta)),e)\qquad\text{for
all}\quad(\delta,\lambda)\in\Gamma.$

This is indeed the case: since $p$ is equivariant for the $\Lambda$-actions,
we have

$p(\beta_{\Delta}(\lambda\delta\lambda^{-1}))=p(\theta_{b}(\lambda)\beta_{\Delta}(\delta))=(\widehat{\alpha}^{\prime}\oplus\theta^{\prime})(\beta_{\Lambda}(\lambda))p(\beta_{\Delta}(\delta)).$

Next, let $\pi$ be a unitary representation of $\Gamma$ on a finite
dimensional vector space $V.$ By Proposition 5, we have to show that there
exists a unitary representation $\widetilde{\pi}$ of $B$ on $V$ such that
$\pi=\widetilde{\pi}\circ\beta$.

Consider a decomposition of $V=V_{1}\oplus\cdots\oplus V_{s}$ into irreducible
$\pi(\Delta)$-invariant subspaces $V_{i}$; denote by
$\sigma_{1},\dots,\sigma_{s}$ the corresponding irreducible representations of
$\Delta.$ By Theorem 1, every $\sigma_{i}$ is of the form
$\sigma_{i}=\chi_{i}\otimes\rho_{i}$ for some $\chi_{i}\in\widehat{\Delta^{\rm
Ab}}$ and $\rho_{i}\in\widehat{\Delta}_{\rm finite}.$

We decompose every $\chi_{i}$ as a product
$\chi_{i}=\chi_{i}^{\prime}\chi_{i}^{\prime\prime}$ with
$\chi_{i}^{\prime}\in\widehat{\Delta^{\rm Ab}}$ of finite order and
$\chi_{i}^{\prime\prime}\in\widehat{\Delta^{\rm Ab}}$ of infinite order. Since
$\chi_{i}^{\prime}$ has finite image, upon replacing $\rho_{i}$ by
$\chi_{i}^{\prime}\otimes\rho_{i}$, we may and will assume that every non
trivial $\chi_{i}$ has infinite order.

Fix $i\in\\{1,\dots,s\\}.$ We can extend $\chi_{i}$ and $\rho_{i}$ to unitary
representations of $\operatorname{Bohr}(\Delta),$ that is, we can find
representations $\widetilde{\chi_{i}}$ and $\widetilde{\rho_{i}}$ of
$\operatorname{Bohr}(\Delta)$ on $V_{i}$ such that
$\chi_{i}=\widetilde{\chi_{i}}\circ\beta_{\Delta}$ and
$\rho_{i}=\widetilde{\rho_{i}}\circ\beta_{\Delta}.$ By Proposition 12, the
stabilizer $\Gamma_{\sigma_{i}}$ of $\sigma_{i}$ has finite index in $\Gamma.$
It follows that the $\Lambda$-orbit of $\sigma_{i}$ is finite, and this
implies that $\chi_{i}\in H;$ hence, $\widetilde{\chi_{i}}$ factorizes through

$C=\operatorname{Bohr}(\Delta^{\rm
Ab})_{0}/(N\cap\operatorname{Bohr}(\Delta^{\rm Ab})_{0})$

and we have $\chi_{i}=\widetilde{\chi_{i}}\circ(p\circ\beta_{\Delta}).$ Since
$\rho_{i}$ has finite image, $\widetilde{\rho_{i}}$ factorizes through
$\operatorname{Prof}(\Delta)$. So,
$\widetilde{\sigma_{i}}:=\widetilde{\chi_{i}}\otimes\widetilde{\rho_{i}}$ is a
unitary representation of $C\times\operatorname{Prof}(\Delta)$ on $V_{i}$. Set

$\widetilde{\pi_{\Delta}}:=\bigoplus_{i=1}^{s}\widetilde{\sigma_{i}}.$

Then $\widetilde{\pi_{\Delta}}$ is a unitary representation of
$C\times\operatorname{Prof}(\Delta)$ on $V$ such that
$\pi|_{\Delta}=\widetilde{\pi_{\Delta}}\circ(\beta|_{\Delta}).$

On the other hand, since $\pi|_{\Lambda}$ is a finite dimensional
representation of $\Lambda,$ we can find a representation
$\widetilde{\pi_{\Lambda}}$ of $\operatorname{Bohr}(\Lambda)$ on $V$ such that
$\pi|_{\Lambda}=\widetilde{\pi_{\Lambda}}\circ(\beta|_{\Lambda}).$ For
$\lambda\in\Lambda$ and $\delta\in\Delta$, we have

$\displaystyle\widetilde{\pi_{\Delta}}(\beta(\lambda)\beta(\delta)\beta(\lambda)^{-1})$
$\displaystyle=\widetilde{\pi_{\Delta}}(\beta(\lambda\delta\lambda)^{-1})$
$\displaystyle=\pi(\lambda\delta\lambda)^{-1})$
$\displaystyle=\pi(\lambda)\pi(\delta)\pi(\lambda)^{-1}$
$\displaystyle=\widetilde{\pi_{\Lambda}}(\beta(\lambda))\widetilde{\pi_{\Delta}}(\beta(\delta))\widetilde{\pi_{\Lambda}}(\beta(\lambda))^{-1}.$

Since $\beta$ has dense image in $B,$ it follows that

$\widetilde{\pi_{\Delta}}(bab^{-1})=\widetilde{\pi_{\Lambda}}(b)\widetilde{\pi_{\Delta}}(a)\widetilde{\pi_{\Lambda}}(b)^{-1}\qquad\text{for
all}\quad(a,b)\in B$

and therefore the formula

$\widetilde{\pi}(a,b)=\widetilde{\pi_{\Delta}}(a)\widetilde{\pi_{\Lambda}}(b)\qquad\text{for
all}\quad(a,b)\in B$

defines a unitary representation of $B$ on $V$ such that
$\pi=\widetilde{\pi}\circ\beta.$

##  5\. Proof of Theorem 3

Recall that we are assuming that $\mathbf{G}$ is a connected, simply-connected
and almost $\mathbf{Q}$-simple algebraic group. The group $\mathbf{G}$ can be
obtained from an absolutely simple algebraic group $\mathbf{H}$ by the so-
called restriction of scalars; more precisely (see [Borel-Tits1, 6.21, (ii)]),
there exists a number field $K$ and an absolutely simple algebraic group
$\mathbf{H}$ over $K$ which is absolutely simple with the following property:
$\mathbf{G}$ can be written as (more precisely, is $\mathbf{Q}$-isomorphic to)
the $\mathbf{Q}$-group
$\mathbf{H}^{\sigma_{1}}\times\dots\times\mathbf{H}^{\sigma_{s}},$ where the
$\sigma_{i}$’s are the different (non conjugate) embeddings of $K$ in
${\mathbf{C}}$. Assuming that $\sigma_{1},\dots,\sigma_{r_{1}}$ are the
embeddings such that $\sigma_{i}(K)\subset{\mathbf{R}},$ we can identify
$\mathbf{G}({\mathbf{R}})$ with

$\mathbf{H}^{\sigma_{1}}({\mathbf{R}})\times\dots\times\mathbf{H}^{\sigma_{r_{1}}}({\mathbf{R}})\times\mathbf{H}^{\sigma_{r_{1}+1}}({\mathbf{C}})\times\dots\times\mathbf{H}^{\sigma_{r_{s}}}({\mathbf{C}}).$

Let $\mathbf{G}_{\rm c}$ be the product of the $\mathbf{H}^{\sigma_{i}}$’s for
which $\mathbf{H}^{\sigma_{i}}({\mathbf{R}})$ is compact.

We assume now that the real semisimple Lie group $\mathbf{G}({\mathbf{R}})$ is
not locally isomorphic to a group of the form $SO(m,1)\times L$ or
$SU(m,1)\times L$ for a compact Lie group $L$. Let
$\Gamma\subset\mathbf{G}(\mathbf{Q})$ be an arithmetic subgroup.

Set $K:=\mathbf{G}_{\rm c}({\mathbf{R}})\times\operatorname{Prof}(\Gamma)$ and
let $\beta:\Gamma\to K$ be defined by
$\beta(\gamma)=(p(\gamma),\alpha(\gamma)),$ where
$p:\mathbf{G}({\mathbf{R}})\to\mathbf{G}_{\rm c}({\mathbf{R}})$ is the
canonical projection and $\alpha:\Gamma\to\operatorname{Prof}(\Gamma)$ the map
associated to $\operatorname{Prof}(\Gamma).$ We claim that $(K,\beta)$ is a
Bohr compactification of $\Gamma$,

First, we show that $\beta(\Gamma)$ has dense image. Observe that
$\mathbf{G}_{\rm c}({\mathbf{R}})$ is connected (see [Borel-Book, (24.6.c)]).
By the Strong Approximation Theorem (see [Platonov, Theorem 7.12]),
$p(\mathbf{G}({\mathbf{Z}}))$ is dense in $\mathbf{G}_{\rm c}({\mathbf{R}})$.
Since $\mathbf{G}_{\rm c}({\mathbf{R}})$ is connected and since $\Gamma$ is
commensurable to $\mathbf{G}({\mathbf{Z}})$, it follows that $p(\Gamma)$ is
dense in $\mathbf{G}_{\rm c}({\mathbf{R}})$. Now, $\alpha(\Gamma)$ is dense in
$\operatorname{Prof}(\Gamma)$ and $\operatorname{Prof}(\Gamma)$ is totally
disconnected. As in the first step of the proof of Theorem 1, we conclude that
$\beta(\Gamma)$ is dense in $K.$

Let $\pi:\Gamma\to U(n)$ be a finite dimensional unitary representation of
$\Gamma.$ Then, by Margulis’ superrigidity theorem (see [Margulis, Chap. VIII,
Theorem B]), [Witte, Corollary 16.4.1]), there exists a continuous
homomorphism $\rho_{1}:\mathbf{G}({\mathbf{R}})\to U(n)$ and a homomorphism
$\rho_{2}:\Gamma\to U(n)$ such that

  * (i)

$\rho_{2}(\Gamma)$ is finite;

  * (ii)

$\rho_{1}(g)\rho_{2}(\gamma)=\rho_{2}(\gamma)\rho_{1}(g)$ for all
$g\in\mathbf{G}({\mathbf{R}})$ and $\gamma\in\Gamma;$

  * (iii)

$\pi(\gamma)=\rho_{1}(\gamma)\rho_{2}(\gamma)$ for all $\gamma\in\Gamma.$

By a classical result of Segal and von Neumann [Segal-vN], $\rho_{1}$
factorizes through $\mathbf{G}_{\rm c}({\mathbf{R}})$, that is,
$\rho_{1}=\rho_{1}^{\prime}\circ p$ for a unitary representation
$\rho_{1}^{\prime}$ of $\mathbf{G}_{\rm c}({\mathbf{R}}).$ It follows from (i)
that $\rho_{2}=\rho^{\prime}_{2}\circ\alpha$ for a unitary representation
$\rho_{2}^{\prime}$ of $\operatorname{Prof}(\Gamma).$ Moreover, (ii) and (iii)
show that $\pi=(\rho_{1}|_{\Gamma})\otimes\rho_{2}.$ Hence,
$\pi=(\rho_{1}^{\prime}\otimes\rho^{\prime}_{2})\circ\otimes\beta$. We
conclude by Proposition 5 that $(K,\beta)$ is a Bohr compactification of
$\Gamma.$

##  6\. A few examples

We compute the Bohr compactification for various examples of arithmetic
groups.

  1. (1)

For an integer $n\geq 1$, the $(2n+1)$-dimensional Heisenberg group is the
unipotent $\mathbf{Q}$-group $\mathbf{H}_{2n+1}$ of matrices of the form

$m(x_{1},\dots,x_{n},y_{1},\dots,y_{n},z):=\left(\begin{array}[]{cccccc}1&x_{1}&\dots&x_{n}&z\\\
0&1&\dots&0&y_{1}\\\ \vdots&\ddots&\ddots&\vdots&\vdots\\\
0&0&\dots&1&y_{n}\\\ 0&0&\dots&0&1\end{array}\right).$

The arithmetic group $\Gamma=\mathbf{H}_{2n+1}({\mathbf{Z}})$ is nilpotent of
step 2; its commutator subgroup $[\Gamma,\Gamma]$ coincides with its center
$\\{m(0,0,z):z\in{\mathbf{Z}}\\}$. So, $\Gamma^{\rm
Ab}\cong{\mathbf{Z}}^{2n}$. We have, by Theorem 1,

$\operatorname{Bohr}(\Gamma)\cong\operatorname{Bohr}({\mathbf{Z}}^{2n})_{0}\times\operatorname{Prof}(\Gamma)$

and hence, by Proposition 11 and Proposition 16

$\operatorname{Bohr}(\Gamma)\cong(\prod_{\omega\in\mathfrak{c}}{\mathbf{A}}/\mathbf{Q})\times\prod_{p\
\text{prime}}\mathbf{H}_{2n+1}({\mathbf{Z}}_{p}).$

  2. (2)

Let $\mathbf{G}=SL_{n}$ for $n\geq 3$ or $\mathbf{G}=Sp_{2n}$ for $n\geq 2.$
Then $SL_{n}({\mathbf{Z}})$ and $Sp_{2n}({\mathbf{Z}})$ are non cocompact
arithmetic lattices in $SL_{n}({\mathbf{R}})$ and $Sp_{2n}({\mathbf{R}}),$
respectively. Hence, we have, by Corollary 4,
$\operatorname{Bohr}(SL_{n}({\mathbf{Z}}))=\operatorname{Prof}(SL_{n}({\mathbf{Z}}))$
and
$\operatorname{Bohr}(Sp_{2n}({\mathbf{Z}}))=\operatorname{Prof}(Sp_{2n}({\mathbf{Z}}))$.
Since $SL_{n}({\mathbf{Z}})$ and $Sp_{2n}({\mathbf{Z}})$ have the congruence
subgroup property, it follows that

$\operatorname{Bohr}(SL_{n}({\mathbf{Z}}))\cong\prod_{p\
\text{prime}}SL_{n}({\mathbf{Z}}_{p})\cong
SL_{n}(\operatorname{Prof}({\mathbf{Z}}))$

and similarly

$\operatorname{Bohr}(Sp_{2n}({\mathbf{Z}}))\cong\prod_{p\
\text{prime}}Sp_{2n}({\mathbf{Z}}_{p})\cong
Sp_{2n}(\operatorname{Prof}({\mathbf{Z}})).$

  3. (3)

The group $\Gamma=SL_{2}({\mathbf{Z}}[\sqrt{2}])$ embeds as a non cocompact
arithmetic lattice of $SL_{2}({\mathbf{R}})\times SL_{2}({\mathbf{R}}).$ So,
by Corollary 4, we have

$\operatorname{Bohr}(SL_{2}({\mathbf{Z}}[\sqrt{2}]))\cong\operatorname{Prof}(SL_{2}({\mathbf{Z}}[\sqrt{2}])).$

Moreover, since $\Gamma$ has the congruence subgroup property (see [Serre,
Corollaire 3]), it follows that

$\operatorname{Bohr}(SL_{2}({\mathbf{Z}}[\sqrt{2}]))\cong\operatorname{Cong}(SL_{2}({\mathbf{Z}}[\sqrt{2}])).$

  4. (4)

For $n\geq 4,$ consider the quadratic form

$q(x_{1},\dots,x_{n})=x_{1}^{2}+\dots+x_{n-1}^{2}-\sqrt{2}x_{n}^{2}-\sqrt{2}x_{n+1}^{2}$

The group $\mathbf{G}=SO(q)$ of unimodular $(n+1)\times(n+1)$-matrices which
preserve $q$ is an almost simple algebraic group over the number field
$\mathbf{Q}[\sqrt{2}].$ The subgroup $\Gamma=SO(q,{\mathbf{Z}}[\sqrt{2}])$ of
${\mathbf{Z}}[\sqrt{2}]$-rational points in $\mathbf{G}$ embeds as a cocompact
lattice of the semisimple real Lie group $SO(n+1)\times SO(n-1,2)$ via the map

$SO(q,\mathbf{Q}[\sqrt{2}])\to SO(n+1)\times
SO(n-1,2),\,\gamma\mapsto(\gamma^{\sigma},\gamma),$

where $\sigma$ is the field automorphism of $\mathbf{Q}[\sqrt{2}]$ given by
$\sigma(\sqrt{2})=-\sqrt{2};$ so, $SO(n+1)\times SO(n-1,2)$ is the group of
real points of the $\mathbf{Q}$-group
$R_{\mathbf{Q}[\sqrt{2}]/\mathbf{Q}}(\mathbf{G})$ obtained by restriction of
scalars from the $\mathbf{Q}[\sqrt{2}]$-group $\mathbf{G}$. Observe that
$R_{\mathbf{Q}[\sqrt{2}]/\mathbf{Q}}(\mathbf{G})$ is almost
$\mathbf{Q}[\sqrt{2}]$-simple since $\mathbf{G}$ is almost
$\mathbf{Q}$-simple. By Theorem 3, we have

$\operatorname{Bohr}(SO(q,{\mathbf{Z}}[\sqrt{2}]))\cong
SO(n+1)\times\operatorname{Prof}(SO(q,{\mathbf{Z}}[\sqrt{2}]).$

  5. (5)

For $d\geq 2,$ let $D$ be a central division algebra over $\mathbf{Q}$ such
that $D\otimes_{\mathbf{Q}}{\mathbf{R}}$ is isomorphic to the algebra
$M_{d}({\mathbf{R}})$ of real $d\times d$-matrices. There exists a subring
$\mathcal{O}$ of $D$ which is a ${\mathbf{Z}}$-lattice in $D$ (a so-called
order in $D$). There is an embedding $\varphi:D\to M_{d}({\mathbf{R}})$ such
that $\varphi(SL_{1}(D)\subset SL_{d}(\mathbf{Q})$ and such that
$\Gamma:=\varphi(SL_{1}(\mathcal{O})$ is an arithmetic cocompact lattice in
$SL_{d}({\mathbf{R}}),$ where $SL_{1}(D)$ is the group of norm one elements in
$D;$ for all this, see [Witte, §6.8.i]. For $d\geq 2,$ we have

$\operatorname{Bohr}(\Gamma)\cong\operatorname{Prof}(\Gamma).$

So, this is an example of a _cocompact_ lattice $\Gamma$ in a simple real Lie
group for which there exists no homomorphism $\Gamma\to U(n)$ with infinite
image; the existence of such examples was mentioned in [Witte, (16.4.3)]

  6. (6)

For $n\geq 3,$ let $\Gamma$ be the semi-direct product
${\mathbf{Z}}^{n}\rtimes SL_{n}({\mathbf{Z}}),$ induced by the usual linear
action of $SL_{n}({\mathbf{Z}})$ on ${\mathbf{R}}^{n}.$ The dual action of
$SL_{n}({\mathbf{Z}})$ on
$\widehat{{\mathbf{Z}}^{n}}\cong{\mathbf{R}}^{n}/{\mathbf{Z}}^{n}$ is given by

$SL_{n}({\mathbf{Z}})\times{\mathbf{R}}^{n}/{\mathbf{Z}}^{n}\to{\mathbf{R}}^{n}/{\mathbf{Z}}^{n},\,(g,x+{\mathbf{Z}}^{n})\mapsto{}^{t}gx+{\mathbf{Z}}^{n}.$

It is well-known and easy to show that the subgroup of
$SL_{n}({\mathbf{Z}})$-periodic orbits in $\widehat{{\mathbf{Z}}^{n}}$
corresponds to $\mathbf{Q}^{n}/{\mathbf{Z}}^{n}$, that is, to the characters
of finite image. It follows from Theorem 2 that

$\operatorname{Bohr}({\mathbf{Z}}^{n}\rtimes
SL_{n}({\mathbf{Z}}))\cong\operatorname{Bohr}(SL_{n}({\mathbf{Z}}))_{0}\times\operatorname{Prof}({\mathbf{Z}}^{n}\rtimes
SL_{n}({\mathbf{Z}})).$

For $n\geq 3,$ we have therefore

$\operatorname{Bohr}({\mathbf{Z}}^{n}\rtimes
SL_{n}({\mathbf{Z}}))\cong\operatorname{Prof}({\mathbf{Z}}^{n}\rtimes
SL_{n}({\mathbf{Z}}))\cong\prod_{p\ \text{prime}}{\mathbf{Z}}_{p}\rtimes
SL_{n}({\mathbf{Z}}_{p}).$

## References

  * [1]
AnzaiHirotadaKakutaniShizuoBohr compactifications of a locally compact abelian
group. iProc. Imp. Acad. Tokyo191943476–480@article{Anzai-Kakutani1, author =
{Anzai, Hirotada}, author = {Kakutani, Shizuo}, title = {Bohr
compactifications of a locally compact Abelian group. I}, journal = {Proc.
Imp. Acad. Tokyo}, volume = {19}, date = {1943}, pages = {476–480}}

  * [3]
BassH.MilnorJ.SerreJ.-P.Solution of the congruence subgroup problem for ${\rm
SL}_{n}\,(n\geq 3)$ and ${\rm Sp}_{2n}\,(n\geq 2)$Inst. Hautes Études Sci.
Publ. Math.33196759–137@article{BMS, author = {Bass, H.}, author = {Milnor,
J.}, author = {Serre, J.-P.}, title = {Solution of the congruence subgroup
problem for ${\rm SL}_{n}\,(n\geq 3)$ and ${\rm Sp}_{2n}\,(n\geq 2)$}, journal
= {Inst. Hautes \'{E}tudes Sci. Publ. Math.}, number = {33}, date = {1967},
pages = {59–137}}

  * [5]
BekkaB.HarpeP. de laValetteA.Kazhdan’s property (t)New Mathematical
Monographs11Cambridge University Press, Cambridge2008@book{BHV, author =
{Bekka, B.}, author = { Harpe, P. de la}, author = {Valette, A.}, title =
{Kazhdan's property (T)}, series = {New Mathematical Monographs}, volume =
{11}, publisher = {Cambridge University Press, Cambridge}, date = {2008}}

  * [7]
BekkaB.HarpeP. de laUnitary representations of groups, duals, and
charactersGraduate Studies in MathematicsAmerican Mathematical Society,
Providence, RI@book{BH, author = {Bekka, B.}, author = { Harpe, P. de la},
title = {Unitary representations of groups, duals, and characters}, series =
{Graduate Studies in Mathematics}, publisher = {American Mathematical Society,
Providence, RI}} BohrH.Zur theorie der fast periodischen funktionenGermanActa
Math.451925129–127@article{Bohr1, author = {Bohr, H.}, title = {Zur Theorie
der fast periodischen Funktionen}, language = {German}, journal = {Acta
Math.}, volume = {45}, date = {1925}, number = {1}, pages = {29–127}}
BohrHaraldZur theorie der fastperiodischen funktionenGermanActa
Math.4619251-2101–214@article{Bohr2, author = {Bohr, Harald}, title = {Zur
Theorie der Fastperiodischen Funktionen}, language = {German}, journal = {Acta
Math.}, volume = {46}, date = {1925}, number = {1-2}, pages = {101–214}}

  * [11]
BorelA.Linear algebraic groupsGraduate Texts in Mathematics126SecondSpringer-
Verlag, New York1991xii+288@book{Borel-Book, author = {Borel, A.}, title =
{Linear algebraic groups}, series = {Graduate Texts in Mathematics}, volume =
{126}, edition = {Second}, publisher = {Springer-Verlag, New York}, year =
{1991}, pages = {xii+288}}

  * [13]
BorelA.Some finiteness properties of adele groups over number fieldsInst.
Hautes Études Sci. Publ. Math.1619635–30@article{Borel, author = {Borel, A.},
title = {Some finiteness properties of adele groups over number fields},
journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.}, number = {16}, date =
{1963}, pages = {5–30}}

  * [15]
BorelA.TitsJ.Groupes réductifsInst. Hautes Études Sci. Publ.
Math.27196555–150@article{Borel-Tits1, author = {Borel, A.}, author = {Tits,
J.}, title = {Groupes r\'{e}ductifs}, journal = {Inst. Hautes \'{E}tudes Sci.
Publ. Math.}, number = {27}, year = {1965}, pages = {55–150}} BorelA.Harish-
ChandraArithmetic subgroups of algebraic groupsAnn. of Math.
(2)751962485–535@article{Borel-HC, author = {Borel, A.}, author = {Harish-
Chandra}, title = {Arithmetic subgroups of algebraic groups}, journal = {Ann.
of Math. (2)}, volume = {75}, date = {1962}, pages = {485–535}}

  * [18]
BourbakiN.Éléments de mathématique. topologie générale. chapitres 1 à
4Hermann, Paris1971xv+357 pp. (not consecutively paged)@book{Bourbaki, author
= {Bourbaki, N.}, title = {\'{E}l\'{e}ments de math\'{e}matique. Topologie
g\'{e}n\'{e}rale. Chapitres 1 \\`a 4}, publisher = {Hermann, Paris}, date =
{1971}, pages = {xv+357 pp. (not consecutively paged)}}

  * [20]
ChevalleyC.Deux théorèmes d’arithmétiqueFrenchJ. Math. Soc.
Japan3195136–44@article{Chevalley, author = {Chevalley, C.}, title = {Deux
th\'{e}or\\`emes d'arithm\'{e}tique}, language = {French}, journal = {J. Math.
Soc. Japan}, volume = {3}, date = {1951}, pages = {36–44}}

  * [22]
DixmierJ.$C^{*}$-algebrasNorth-Holland Publishing Co., Amsterdam-New York-
Oxford1977@book{Dixm–C*, author = {Dixmier, J.}, title = {{$C\sp*$}-algebras},
publisher = {North-Holland Publishing Co., Amsterdam-New York-Oxford}, year =
{1977}}

  * [24]
DruţuC.KapovichM.Geometric group theoryAmerican Mathematical Society
Colloquium Publications63American Mathematical Society, Providence,
RI2018xx+819@book{Drutu, author = {Dru\c{t}u, C.}, author = {Kapovich, M.},
title = {Geometric group theory}, series = {American Mathematical Society
Colloquium Publications}, volume = {63}, publisher = {American Mathematical
Society, Providence, RI}, date = {2018}, pages = {xx+819}}
FranksJ.HandelM.Distortion elements in group actions on surfacesDuke Math.
J.13120063441–468@article{Franks, author = {Franks, J.}, author = {Handel,
M.}, title = {Distortion elements in group actions on surfaces}, journal =
{Duke Math. J.}, volume = {131}, date = {2006}, number = {3}, pages =
{441–468}}

  * [27]
GerstenS. M.ShortH. B.Rational subgroups of biautomatic groupsAnn. of Math.
(2)13419911125–158@article{Gersten, author = {Gersten, S. M.}, author =
{Short, H. B.}, title = {Rational subgroups of biautomatic groups}, journal =
{Ann. of Math. (2)}, volume = {134}, date = {1991}, number = {1}, pages =
{125–158}} BallmannW.GromovM.SchroederV.Manifolds of nonpositive
curvatureProgress in Mathematics61Birkhäuser Boston, Inc., Boston,
MA1985vi+263HartJ. E.KunenK.Bohr compactifications of non-abelian
groupsProceedings of the 16th Summer Conference on General Topology and its
Applications (New York)Topology Proc.262001/022593–626@article{Hart-Kunen,
author = {Hart, J. E.}, author = {Kunen, K.}, title = {Bohr compactifications
of non-abelian groups}, booktitle = {Proceedings of the 16th Summer Conference
on General Topology and its Applications (New York)}, journal = {Topology
Proc.}, volume = {26}, date = {2001/02}, number = {2}, pages = {593–626}}
HewittE.RossK. A.Abstract harmonic analysis. vol. i1152Springer-Verlag,
Berlin-New York1979ix+519@book{HewittRoss, author = {Hewitt, E.}, author =
{Ross, K. A.}, title = {Abstract harmonic analysis. Vol. I}, volume = {115},
edition = {2}, publisher = {Springer-Verlag, Berlin-New York}, date = {1979},
pages = {ix+519}}

  * [32]
HolmP.On the bohr compactificationMath. Ann.156196434–46@article{Holm, author
= {Holm, P.}, title = {On the Bohr compactification}, journal = {Math. Ann.},
volume = {156}, date = {1964}, pages = {34–46}}

  * [34]
LubotzkyA.MozesS.RaghunathanM. S.The word and riemannian metrics on lattices
of semisimple groupsInst. Hautes Études Sci. Publ. Math.9120005–53
(2001)@article{LMR, author = {Lubotzky, A.}, author = {Mozes, S.}, author =
{Raghunathan, M. S.}, title = {The word and Riemannian metrics on lattices of
semisimple groups}, journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
number = {91}, date = {2000}, pages = {5–53 (2001)}}

  * [36]
MargulisG. A.Discrete subgroups of semisimple lie groupsSpringer-Verlag,
Berlin1991x+388Review MathReviews@book{Margulis, author = {Margulis, G. A.},
title = {Discrete subgroups of semisimple Lie groups}, publisher = {Springer-
Verlag, Berlin}, date = {1991}, pages = {x+388}, review = {\MR{1090825}}}
MostowG. D.Fully reducible subgroups of algebraic groupsAmer. J. Math.American
Journal of Mathematics781956200–221@article{Mostow, author = {Mostow, G. D.},
title = {Fully reducible subgroups of algebraic groups}, journal = {Amer. J.
Math.}, fjournal = {American Journal of Mathematics}, volume = {78}, year =
{1956}, pages = {200–221}}

  * [39]
v. NeumannJ.Almost periodic functions in a group. iTrans. Amer. Math.
Soc.3619343445–492ISSN 0002-9947Review MathReviews@article{VN, author = {v.
Neumann, J.}, title = {Almost periodic functions in a group. I}, journal =
{Trans. Amer. Math. Soc.}, volume = {36}, date = {1934}, number = {3}, pages =
{445–492}, issn = {0002-9947}, review = {\MR{1501752}}}

  * [41]
PlatonovV.RapinchukA.Algebraic groups and number theoryPure and Applied
Mathematics139Academic Press, Inc., Boston, MA1994xii+614@book{Platonov,
author = {Platonov, V. }, author = {Rapinchuk, A.}, title = {Algebraic groups
and number theory}, series = {Pure and Applied Mathematics}, volume = {139},
publisher = {Academic Press, Inc., Boston, MA}, year = {1994}, pages =
{xii+614}} RaghunathanM. S.On the congruence subgroup problemInst. Hautes
Études Sci. Publ. Math.461976107–161@article{Raghunathan-CSP, author =
{Raghunathan, M. S.}, title = {On the congruence subgroup problem}, journal =
{Inst. Hautes \'{E}tudes Sci. Publ. Math.}, number = {46}, date = {1976},
pages = {107–161}}

  * [44]
RaghunathanM. S.Discrete subgroups of lie groupsSpringer-Verlag, New York-
Heidelberg1972@book{Raghunathan, author = {Raghunathan, M. S.}, title =
{Discrete subgroups of Lie groups}, publisher = {Springer-Verlag, New York-
Heidelberg}, date = {1972}}

  * [46]
RibesL.ZalesskiiP.Profinite groups40Springer-Verlag,
Berlin2000xiv+435@book{Ribes-Zal, author = {Ribes, L.}, author = {Zalesskii,
P.}, title = {Profinite groups}, volume = {40}, publisher = {Springer-Verlag,
Berlin}, date = {2000}, pages = {xiv+435}}

  * [48]
SegalI. E.von NeumannJ.A theorem on unitary representations of semisimple lie
groupsAnn. of Math. (2)521950509–517@article{Segal-vN, author = {Segal, I.
E.}, author = {von Neumann, J.}, title = {A theorem on unitary representations
of semisimple Lie groups}, journal = {Ann. of Math. (2)}, volume = {52}, date
= {1950}, pages = {509–517}} SerreJ.-P.Le problème des groupes de congruence
pour sl2FrenchAnn. of Math. (2)921970489–527@article{Serre, author = {Serre,
J.-P.}, title = {Le probl\\`eme des groupes de congruence pour SL2}, language
= {French}, journal = {Ann. of Math. (2)}, volume = {92}, date = {1970}, pages
= {489–527}}

  * [51]
TitsJ.Free subgroups in linear groupsJ. Algebra201972250–270@article{Tits,
author = {Tits, J.}, title = {Free subgroups in linear groups}, journal = {J.
Algebra}, volume = {20}, date = {1972}, pages = {250–270}}

  * [53]
TitsJacquesGroupes à croissance polynomiale (d’après m. gromov et
al.)title={Bourbaki Seminar, Vol. 1980/81}, series={Lecture Notes in Math.},
volume={901}, publisher={Springer, Berlin-New York},
1981176–188@article{Tits2, author = {Tits, Jacques}, title = {Groupes \\`a
croissance polynomiale (d'apr\\`es M. Gromov et al.)}, conference =
{title={Bourbaki Seminar, Vol. 1980/81}, }, book = {series={Lecture Notes in
Math.}, volume={901}, publisher={Springer, Berlin-New York}, }, date = {1981},
pages = {176–188}}

  * [55]
WehrfritzB. A. F.Infinite linear groups. an account of the group-theoretic
properties of infinite groups of matricesSpringer-Verlag, New York-
Heidelberg1973xiv+229@book{Wehrfritz, author = {Wehrfritz, B. A. F.}, title =
{Infinite linear groups. An account of the group-theoretic properties of
infinite groups of matrices}, publisher = {Springer-Verlag, New York-
Heidelberg}, date = {1973}, pages = {xiv+229}}

  * [57]
WeilA.L’intégration dans les groupes topologiques et ses applicationsHermann &
Cie, Paris1940158@book{Weil, author = {Weil, A.}, title = {L'int\'{e}gration
dans les groupes topologiques et ses applications}, publisher = {Hermann \&
Cie, Paris}, date = {1940}, pages = {158}}

  * [59]
MorrisD. WitteIntroduction to arithmetic groupsDeductive
Press2015xii+475@book{Witte, author = {Morris, D. Witte}, title =
{Introduction to arithmetic groups}, publisher = {Deductive Press}, date =
{2015}, pages = {xii+475}}

  * [61]