# Charmenability and Stiffness of Arithmetic Groups

Uri Bader, Itamar Vigdorovich

###### Abstract

We characterize charmenability among arithmetic groups and deduce dichotomy
statements pertaining normal subgroups, characters, dynamics, representations
and associated operator algebras. We do this by studying the stationary
dynamics on the space of characters of the amenable radical, and in particular
we establish stiffness: any stationary probability measure is invariant. This
generalizes a classical result of Furstenberg for dynamics on the torus. Under
a higher rank assumption, we show that any action on the space of characters
of a finitely generated virtually nilpotent group is stiff.

##  1 Introduction

In this paper we are concerned with the dynamics of the conjugation action of
a group $\Lambda$ on the compact convex space $\operatorname{PD}_{1}(\Lambda)$
consisting all normalized positive-definite functions on $\Lambda$. In
particular, we are interested in the structure of invariant compact convex
subsets and faces of $\operatorname{PD}_{1}(\Lambda)$. A special attention
will be given to the subset of fixed points
$\operatorname{Tr}(\Lambda):=\operatorname{PD}_{1}(\Lambda)^{\Lambda}$, and
the corresponding set of extreme points
$\operatorname{Ch}(\Lambda):=\operatorname{ext}(\operatorname{Tr}(\Lambda))$.
An element of $\operatorname{Tr}(\Lambda)$ will be called a _trace_ of
$\Lambda$ and an element of $\operatorname{Ch}(\Lambda)$ will be called a
_character_ 111Note that in some texts, e.g [BBHP22], the term character
refers to what we call here a trace..

######  Definition 1.1.

A countable group $\Lambda$ is said to be _charmenable_ if it satisfies the
following two properties:

  1. 1.

(Ubiquity of traces) Every compact convex $\Lambda$-invariant subset of
$\operatorname{PD}_{1}(\Lambda)$ contains a trace.

  2. 2.

(Character dichotomy) Every character of $\Lambda$ is either supported on
$\operatorname{Rad}(\Lambda)$ or it is von Neumann amenable (see Definition
3.10).

Here $\operatorname{Rad}(\Lambda)$ denotes the _amenable radical_ of
$\Lambda$, namely its unique maximal normal amenable subgroup, and an element
of $\operatorname{PD}_{1}(\Lambda)$ is supported on
$\operatorname{Rad}(\Lambda)$ if it vanishes outside this subgroup. We note
that amenable groups are charmenable in a strong way: they satisfy both
conditions in item 2 above. For non-amenable groups these two conditions are
exclusive.

Our main result is the characterization of charmenability among arithmetic
groups, which in this paper are defined as follows.

######  Definition 1.2.

Given a $\mathbb{Q}$-algebraic group $\mathbf{G}$, a subgroup
$\Lambda<\mathbf{G}(\mathbb{Q})$ is said to be an _arithmetic subgroup of
$\mathbf{G}$_ if for some (equivalently, any) faithful $\mathbb{Q}$-rational
representation $\mathbf{G}\to\operatorname{GL}_{n}$, $\Lambda$ is
commensurable with the preimage of $\operatorname{GL}_{n}(\mathbb{Z})$ in
$\mathbf{G}(\mathbb{Q})$.

######  Theorem A.

Let $\mathbf{G}$ be a connected $\mathbb{Q}$-algebraic group. Then either all
arithmetic subgroups of $\mathbf{G}$ are charmenable or none of them are.
Denoting by $\mathbf{R}$ the solvable radical of $\mathbf{G}$, the first case
occurs if and only if the real rank of $\mathbf{G}/\mathbf{R}$ is not equal to
$1$ and $\mathbf{G}/\mathbf{R}$ has at most one $\mathbb{R}$-isotropic
$\mathbb{Q}$-simple factor.

The following proposition describes a variety of properties that charmenable
groups satisfy, most of which are proved in [BBHP22]. An exception is the
third property which is novel (see Theorem 6.6). The notation involved will be
clarified and its proof will be discussed in §6.

######  Proposition 1.3.

For a charmenable group $\Lambda$ the following hold:

  1. 1.

Every normal subgroup is either amenable or co-amenable.

  2. 2.

Every trace is either amenable or supported on $\operatorname{Rad}(\Lambda)$.

  3. 3.

For every probability measure preserving action with a spectral gap, the point
stabilizers are either a.e amenable or a.e co-amenable.

  4. 4.

Every URS is either contained in
$\operatorname{Sub}(\operatorname{Rad}(\Lambda))$ or it admits a
$\Lambda$-invariant probability measure (in which case the previous item
applies).

  5. 5.

Every unitary representation is either amenable or weakly contains a
representation which is induced from $\operatorname{Rad}(\Lambda)$.

  6. 6.

If $M$ is a finite factor von Neumann algebra admitting a representation
$\Lambda\to\mathcal{U}(M)$ with ultraweak-dense image, then either $M$ is
hyperfinite, or $M$ is isomorphic to the von Neumann algebra of a
representation which is induced from $\operatorname{Rad}(\Lambda)$.

  7. 7.

The $C^{*}$-algebra $A$ of any non-amenable unitary representation of
$\Lambda$ admits a unital surjective *-homomorphism onto the $C^{*}$-algebra
of a representation which is induced from $\operatorname{Rad}(\Lambda)$.
Furthermore, any maximal ideal of $A$ is the kernel of such a surjection, and
any tracial state on $A$ factorizes through such a surjection.

When property (T) is assumed, stronger statements are obtained. For example,
amenable (co-amenable) subgroups must be finite (of finite index), amenable
representations must contain a finite dimensional subrepresentation, etc. With
further assumptions on the amenable radical we obtain:

######  Proposition 1.4.

Let $\Lambda$ be a charmenable group with property (T). Assume that
$\operatorname{Rad}(\Lambda)$ has countably many subgroups. Then every ergodic
IRS is finitely supported and every URS is finite.

The amenable radical of an arithmetic group is virtually polycyclic (Corollary
4.10) and thus has countably many subgroups. We therefore deduce the following
corollary.

######  Corollary B.

Let $\mathbf{G}$ be a connected $\mathbb{Q}$-algebraic group and let
$\Lambda<\mathbf{G}(\mathbb{Q})$ be an arithmetic subgroup. Denoting by
$\mathbf{R}$ the solvable radical of $\mathbf{G}$, if the real rank of
$\mathbf{G}/\mathbf{R}$ is not equal to $1$ and $\mathbf{G}/\mathbf{R}$ has at
most one $\mathbb{R}$-isotropic $\mathbb{Q}$-simple factor then $\Lambda$
satisfies all properties in Proposition 1.3. If moreover $\Lambda$ has
property (T), then every ergodic IRS is finitely supported and every URS is
finite.

The proof of Theorem A will be given in §5.2 and the proof of Propositions 1.3
and 1.4 are concluded in §6. Theorem A in case $\mathbf{R}$ is trivial, namely
when $\mathbf{G}$ is semisimple, was proved in [BH21] and [BBHP22]. Our main
concern in this paper is dealing with the case where $\mathbf{R}$ is non-
trivial. In that case [BBHP22, Proposition 4.19] enables us to reduce the
proof of Theorem A to a certain stiffness result. A prototypical example is
the proof of [BBHP22, Theorem C] which establishes the charmenability of
$\operatorname{SL}_{n}(\mathbb{Z})\ltimes\mathbb{Z}^{n}$ using Furstenberg’s
classical theorem [Fur98] regarding the stiffness of the standard action of
$\operatorname{SL}_{n}(\mathbb{Z})$ on $\mathbb{T}^{n}$ . The stiffness result
that we need is Theorem C below. In order to state it, we need some
preparation.

######  Definition 1.5.

Let $\Lambda$ be a countable group acting on a topological space $X$ by
homeomorphisms. Given a probability measure $\mu$ on $\Lambda$, we say that
the $\Lambda$-action on $X$ is _$\mu$ -stiff_ if any $\mu$-stationary
probability measure222In this paper, all measures on topological spaces are
assumed to be Borel. $\nu$ on $X$, that is $\nu$ for which
$\mu*\nu:=\sum_{g\in\Lambda}\mu(g)g.\nu=\nu$, is $\Lambda$-invariant.

We note that since we are not assuming $X$ to be compact, there may be no
$\mu$-stationary measures at all on $X$, in which case the action is stiff in
a vacuous sense. This is never the case in the type of dynamical systems we
consider.

######  Proposition 1.6.

Let $\Lambda$ be a countable group acting by automorphisms on a countable
group $\Gamma$. Then for every probability measure $\mu$ on $\Gamma$, the set
of $\mu$-stationary probability measures on $\operatorname{Ch}(\Gamma)$ is
non-empty.

Note that $\operatorname{Ch}(\Gamma)$ is Polish, but may be non-compact (see
Example 1.9 bellow). The proof of Proposition 1.6 will be given in §3.1.

A general group has no canonical probability measure, and stiffness may very
well depend on different choices of $\mu$. However, for arithmetic groups we
have preferred measures.

######  Definition 1.7.

Let $\mathbf{G}$ be an $\mathbb{R}$-algebraic group, $\mathbf{P}<\mathbf{G}$
be a minimal $\mathbb{R}$-parabolic subgroup and $\mathbf{N}<\mathbf{G}$ be
the normalizer of $\mathbf{P}$ (note that $\mathbf{P}$ contains the solvable
radical of $\mathbf{G}$, $\mathbf{P}<\mathbf{N}$ is of finite index and if
$\mathbf{G}$ is connected then $\mathbf{P}=\mathbf{N}$). A probability measure
$\mu$ on $\mathbf{G}(\mathbb{R})$ (not necessarily fully supported) is called
a _Furstenberg measure_ if in the $\mathbf{G}(\mathbb{R})$-invariant measure
class on $\mathbf{G}(\mathbb{R})/\mathbf{N}(\mathbb{R})$ there exists a
probability measure with which this space is a Furstenberg-Poisson boundary
for $(\mathbf{G}(\mathbb{R}),\mu)$. If $\Lambda<\mathbf{G}(\mathbb{R})$ is a
closed subgroup which is generated as a semigroup by the support of the
Furstenberg measure $\mu$, we will say that $\mu$ is a _Furstenberg measure on
$\Lambda$_.

Furstenberg proved that lattices in semisimple groups carry Furstenberg
measures, see Proposition 2.24 for an elaboration. The following is a
corollary of the above fact.

######  Proposition 1.8.

For every $\mathbb{Q}$-algebraic group $\mathbf{G}$ and an arithmetic subgroup
$\Lambda$, there exists a Furstenberg measure on $\Lambda$.

The proof of Proposition 1.8 will be given in §2.6. We are now ready to state
our main stiffness result.

######  Theorem C.

For an arithmetic group $\Lambda$ endowed with a Furstenberg measure $\mu$,
the action of $\Lambda$ on $\operatorname{Ch}(\operatorname{Rad}(\Lambda))$ is
$\mu$-stiff.

Note that this theorem, unlike Theorems A and B, involves no rank assumption.
A main ingredient in the proof of Theorem C is Theorem 4.1 which regards
stiffness of the action of lattices in a semisimple Lie group on the space of
characters of a nilpotent group. Another consequence of the latter theorem,
which might be of independent interest, is the following.

######  Theorem D.

Let $\Lambda$ be an irreducible lattice in a connected semisimple Lie group
$G$ with finite center and let $\mu$ be a Furstenberg measure on $\Lambda$.
Assume that $G$ is not isogenous to a group of the form
$\operatorname{SO}(1,m)\times K$ or $\operatorname{SU}(1,m)\times K$, with $K$
compact. Then for any action of $\Lambda$ by automorphisms on a finitely
generated virtually nilpotent group $\Gamma$, the corresponding action on
$\operatorname{Ch}(\Gamma)$ is $\mu$-stiff.

The proofs of Theorem C, Theorem D and Theorem 4.1 will be given in §4. If
$\Gamma$ is a finitely generated abelian group then
$\operatorname{Ch}(\Gamma)$, which coincides with $\hat{\Gamma}$, the
Pontryagin dual of $\Gamma$, is a finite union of tori. However, Theorems C, D
and 4.1 are not to be confused with the results of Benoist-Quint regarding
stiffness for actions on nilmanifolds [BQ13]. Indeed, the space
$\operatorname{Ch}(\Gamma)$ for a finitely generated virtually nilpotent group
which is not virtually abelian is neither homogeneous nor a manifold.

######  Example 1.9.

The space of characters $\operatorname{Ch}(H)$ of the discrete Heisenberg
group

$H=\left\\{\left(\begin{array}[]{ccc}1&x&z\\\ 0&1&y\\\
0&0&1\end{array}\right):x,y,z\in\mathbb{Z}\right\\}$

is homeomorphic to the subset of $\mathbb{R}^{3}$ illustrated in the following
picture:

Here points on the horizontal circle represents the space of characters of the
center $Z<H$. The inclusion $Z\subseteq H$ induces a surjective continuous map
$\operatorname{Ch}(H)\to\operatorname{Ch}(Z)\simeq\hat{Z}\cong\mathbb{R}/\mathbb{Z}$
(see Lemma 3.12). The fiber over an irrational point of
$\mathbb{R}/\mathbb{Z}$ consists of a single point, whereas the fiber of a
reduced fraction $\frac{p}{q}\in\mathbb{R}/\mathbb{Z}$ could be identified
with $\mathbb{R}^{2}/q^{-1}\mathbb{Z}^{2}$, the Pontryagin dual of
$q\mathbb{Z}^{2}$ which could be identified with the center of $H/qZ$ (see
Theorem 4.6). We note that $\operatorname{Ch}(H)$ is not compact: the rational
points of the horizontal circle in the picture above are excluded.

###  1.1 Related works

An excellent introduction to the theory of characters is [BdlH19]. A
comprehensive study of characters of nilpotent groups was carried since the
1970’s, see e.g [How77, Kan80, CM84, Kan06]. This study was recently
generalized to polycyclic groups, and thus to arithmetic subgroups of solvable
algebraic groups [LV22].

A classification of the characters of arithmetic subgroups of higher rank
semisimple algebraic groups was completed in [Pet14] following [Bek07]. The
situation in the rank one case is complicated and somewhat less understood,
see e.g [OSV]. We know of no reference dealing with character theory of
general arithmetic groups, though the paper [BF20] is closely related.

Dynamical aspects of spaces of positive-definite functions on lattices in
semisimple groups were considered recently in [BH21], [BBHP22] and [BBH21],
see also [Hou21]. The notion of “charmenability” is highlighted in [BBHP22].
Dynamical questions regarding spaces of measures on tori are classical. In
particular, the question of “stiffness” is dealt with in [BFLM11] and [BQ13],
following the pioneering work [Fur98]. Surprisingly, it seems that questions
regarding dynamics on the spaces of characters of non-abelian nilpotent groups
were never studied before. For our main goal in this paper, proving Theorem A,
the stiffness result of Theorem C is sufficient. However, it seems to us that
the latter theorem is not optimal, as we deal only with Furstenberg measures,
and we hope to improve it in a future work. Another aspect in which we hope to
improve this work in the future is by considering $S$-arithmetic groups. This
will force us to tackle characters on nilpotent groups which are not finitely
generated, but we do not foresee a major problem in doing so.

###  1.2 The structure of this paper

Sections §2 and §3 are devoted to introducing the needed preliminaries and
basic results regarding stationary dynamics and characters correspondingly. In
§4 we state and prove Theorem 4.1 and use it to prove Theorems C and D. In §5
we prove Theorem A. §6 is devoted to proving Propositions 1.3 and 1.4.

###  1.3 Acknowledgments

This work was supported by the ISF Moked 713510 grant number 2919/19. The
second author would like thank Barak Weiss for useful discussions and
references. We are grateful to Tamuz Ofer for the figure in Example 1.9. Both
authors thank the members of the Midrasha on Groups at the Weizmann Institute
for their support, friendship and professional encouragement.

##  2 Stationary dynamics

In this section we define and discuss general statements regarding stationary
dynamics associated to a group $G$. For almost all of our purposes it suffices
to consider the case where $G$ is countable. However, as some of our claims
might be of independent interest, we shall mostly work in the setting where
$G$ is a general locally compact second countable group.

###  2.1 Ergodic theoretical preliminaries

By a _Borel space_ we mean a set endowed with a $\sigma$-algebra. Morphisms
between Borel spaces are just measurable maps, and they are called _Borel
maps_. A _standard Borel space_ is a Borel space which is isomorphic as such
to the underlying Borel space of a Polish space. By a _Lebesgue space_ we
understand a standard Borel space endowed with a measure class. A _Lebesgue
map_ from a Lebesgue space to a Borel space is a class of a.e equal Borel
maps, defined on a conull subset of the domain. Morphisms between Lebesgue
spaces are measure class preserving Lebesgue maps. We note that the category
of Lebesgue spaces is equivalent to the opposite category of abelian von
Neumann subalgebras of bounded operators on separable Hilbert spaces, by the
functor $X\mapsto L^{\infty}(X)\subseteq\mathcal{B}(L^{2}(X))$.

Let $G$ be a locally compact second countable group. An action of $G$ on a
Borel space is _Borel_ if the corresponding action map is Borel. A _Borel
$G$-space_ is a Borel space endowed with a Borel action of $G$. A _Lebesgue
$G$-space_ is a Lebesgue space endowed with a Borel $G$-action that preserves
the measure class. The Lebesgue $G$-space is called _ergodic_ if every
measurable $G$-invariant subset is either null or conull.

For a topological space $X$ we denote by $\operatorname{Prob}(X)$ the space of
all Radon probability measures endowed with the weak-* topology. If $X$ is
Polish (resp. compact) then so is $\operatorname{Prob}(X)$, and all Borel
probability measures are Radon. Suppose $X$ and $Y$ are Polish spaces and that
$p:X\to Y$ is a continuous surjective map. Let $\eta\in\operatorname{Prob}(Y)$
and consider a Lebesgue map $Y\to\operatorname{Prob}(X)$ that takes $y$ to a
measure $\nu_{y}$ supported on the fiber $p^{-1}(y)$. Then one may integrate
these $\nu_{y}$ into a probability measure $\nu=\int_{Y}\nu_{y}d\eta$. That
is, for a bounded continuous function $f$ on $X$, $\int_{X}f\,{\rm
d}\nu=\int_{Y}\left(\int_{p^{-1}(y)}fd\nu_{y}\right)\,{\rm d}\eta$. The
_disintegration theorem_ states that every $\nu\in\operatorname{Prob}(X)$ is
given by an integration as above w.r.t a unique Lebesgue map
$Y\to\operatorname{Prob}(X):y\mapsto\nu_{y}$. This is called the
_disintegration of $\nu$ along $p$_.

Suppose $X$ and $Y$ are Polish $G$-spaces and that $p:X\to Y$ is a continuous
and surjective $G$-equivariant map. Consider $\nu\in\operatorname{Prob}(X)$
whose measure class is preserved by $G$, and the pushforward measure
$p_{*}\nu$ on $Y$. Then $X$ is said to be _measure preserving relative to $Y$_
if the disintegration map $Y\to\operatorname{Prob}(X)$ associated with $\nu$
is a $G$-equivariant.

###  2.2 Choquet simplices and stationary dynamics

A general reference for Choquet Theory is [Phe01]. Recall that any compact
convex set $C$ in a locally convex topological vector space is the closed
convex hull of its extreme points $\operatorname{Ext}(C)$, by the Krein-Milman
theorem. Assuming that $C$ is metrizable, $\operatorname{Ext}(C)$ is a
$G_{\delta}$ subset of $C$, thus a Polish space. For a probability measure
$\nu$ on $\operatorname{Ext}(C)$ we define the barycenter,
$\operatorname{bar}(\nu)$, to be the unique point $c\in C$ such that
$\phi(c)=\int\phi\,{\rm d}\nu$ for every continuous linear functional on the
ambient space. We obtain an affine continuous map
$\operatorname{bar}:\operatorname{Prob}(\operatorname{Ext}(C))\to C$.
Choquet’s theorem says that this map is surjective.

######  Definition 2.1.

The metrizable compact convex set $C$ is said to be a _Choquet simplex_ if the
map $\operatorname{bar}$ is bijective.

For a metrizable Choquet simplex $C$, the space
$\operatorname{Prob}(\operatorname{Ext}(C))$ is a Polish space, thus the map
$\operatorname{bar}$ is a Borel isomorphism by Suslin’s theorem. It thus makes
sense to consider the category whose objects are Polish convex spaces and its
morphisms are Borel affine maps. The space $\operatorname{Prob}(X)$, for a
Polish space $X$, is a prominent example and a Polish convex space is called a
_Polish simplex_ if it is Borel affine isomorphic to a space of the form
$\operatorname{Prob}(X)$. Note that a Polish simplex is a Choquet simplex iff
it is compact.

Given a Polish space $X$ and a continuous action of topological group $G$ by
homeomorphisms, we have the standard identification
$\operatorname{Erg}_{G}(X)=\operatorname{Ext}(\operatorname{Prob}(X)^{G})$,
where $\operatorname{Erg}_{G}(X)$ denote the space of ergodic $G$-invariant
probability measures on $X$. Integration gives a continuous affine map
$\operatorname{Prob}(\operatorname{Erg}_{G}(X))\to\operatorname{Prob}(X)^{G}$,
and this map is a bijection by the existence and uniqueness of the
disintegration map, considering for a given $\nu\in\operatorname{Prob}(X)^{G}$
the $\nu$-a.e defined Borel map dualizing the von Neumann algebra inclusion
$L^{\infty}(X,\nu)^{G}<L^{\infty}(X,\nu)$. Thus $\operatorname{Prob}(X)^{G}$
is a Polish simplex.

######  Proposition 2.2.

Let $C$ be a metrizable Choquet simplex and assume the group $G$ acts
continuously on $C$ by affine transformations. then $C^{G}$ is a metrizable
Choquet simplex.

###### Proof.

Denote $X=\operatorname{Ext}(C)$. Then the Borel affine $G$-equivariant
isomorphism $\operatorname{bar}:\operatorname{Prob}(X)\to C$ induces a Borel
affine isomorphism $\operatorname{Prob}(X)^{G}\to C^{G}$. Since
$\operatorname{Prob}(X)^{G}$ is a Polish simplex, it follows that $C^{G}$ is a
Polish simplex. It is a Choquet simplex as it is compact. ∎

For another proof of Proposition 2.2, see [KS21, Corrolary 12.13].

Next, we provide an analogue which regards stationary measures. We start by
giving some background. Consider a continuous (strong operator topology)
action of locally compact second countable group $G$ on a Banach space $E$ by
linear isometries. This induces a continuous action on the dual space $E^{*}$
endowed with the weak-* topology. We define the convolution between a
probability measure $\mu$ on $G$ and an element $\varphi\in E^{*}$ by:

$\displaystyle(\mu*\varphi)(v):=\int_{G}\varphi(g^{-1}.v)\,{\rm d}\mu(g)\hskip
20.0pt(v\in E)$ (2.1)

It is easy to see that this integral is absolutely convergent, and that
$\mu*\varphi$ is again an element of $E^{*}$. We say that $\varphi$ is _$\mu$
-stationary_ if $\mu*\varphi=\varphi$. Equation 2.1 thus defines a continuous
action of the topological semigroup $\operatorname{Prob}(G)$, with the
convolution operation and the weak-* topology, on $E^{*}$. We are interested
in $\operatorname{Prob}(G)$-invariant subsets $C\subseteq E^{*}$. If $C$ is
convex, weak-* compact and $G$-invariant, then $C$ is definitely
$\operatorname{Prob}(G)$-invariant. Indeed the convolution operation is given
by the following composition:

$\operatorname{Prob}(G)\times C\to\operatorname{Prob}(G\times
C)\to\operatorname{Prob}(C)\to
C,\quad(\mu,c)\mapsto(\mu\times\delta_{c})\mapsto
a_{*}(\mu\times\delta_{c})\mapsto\operatorname{bar}(a_{*}(\mu\times\delta_{c})).$

where $a:G\times C\to C$ is the action map. Another important example is when
$C=\operatorname{Prob}(X)$ where $X$ is a Polish space and the action on $C$
comes from a continuous action of $G$ on $X$. In that case $C$ is a convex
subset of the dual of the Banach space of all bounded continuous functions on
$X$ and the convolution operation is given by the pushforward of the action
map:

$\operatorname{Prob}(G)\times\operatorname{Prob}(X)\hookrightarrow\operatorname{Prob}(G\times
X)\to\operatorname{Prob}(X)$

######  Definition 2.3.

Let $G$ be a locally compact group. $\mu\in\operatorname{Prob}(G)$ is said to
be _admissible_ if it is absolutely continuous with respect to the Haar
measure class on $G$, and it is not supported on a closed sub-semigroup of
$G$.

It is well known and easy to prove that, for $\mu$ admissible, any
$\mu$-stationary probability measure has a $G$-invariant class.

######  Lemma 2.4.

Let $X$ be a Polish space and $G$ a locally compact second countable group
that acts on it continuously by homeomorphisms. Fix an admissible probability
measure $\mu$ on $G$. Then a $\mu$-stationary probability measure $\nu$ on $X$
is ergodic if and only if it is an extreme point of
$\operatorname{Prob}(X)^{\mu}$.

###### Proof.

This is [BS06, Corollary 2.7]. In this reference, $X$ is assumed compact, but
the proof works just as well if $X$ is Polish. ∎

We denote by $\operatorname{Erg}_{\mu}(X)\subset\operatorname{Prob}(X)^{\mu}$
the subset of ergodic measures. In view of Lemma 2.4, we have that the
integration map
$\operatorname{Prob}(\operatorname{Erg}_{\mu}(X))\to\operatorname{Prob}(X)^{\mu}$,
establishes a continuous affine bijection, again by the existence and
uniqueness of the disintegration map, considering for a given
$\nu\in\operatorname{Prob}(X)^{\mu}$ the $\nu$-a.e defined Borel map dualizing
the von Neumann algebra inclusion $L^{\infty}(X,\nu)^{\mu}<L^{\infty}(X,\nu)$.
Thus $\operatorname{Prob}(X)^{\mu}$ is a Polish simplex.

######  Lemma 2.5.

Let $G$ act on a metrizable Choquet simplex $C$ continuously by affine
homeomorphisms. Let $X$ denote the set of extreme points of $C$ and consider
the induced action of $\operatorname{Prob}(G)$ on $\operatorname{Prob}(X)$.
Then the barycenter map $\operatorname{Prob}(X)\to C$ is a
$\operatorname{Prob}(G)$-equivariant continuous bijection and thus induces a
continuous bijection of non-empty spaces $\operatorname{Prob}(X)^{\mu}\to
C^{\mu}$, for any $\mu\in\operatorname{Prob}(G)$.

###### Proof.

The map $\operatorname{Prob}(X)\to C$ is a continuous affine bijection. It is
obviously $G$-equivariant, thus also $\mu$-equivariant for any
$\mu\in\operatorname{Prob}(G)$. Therefore it induces a bijection
$\operatorname{Prob}(X)^{\mu}\to C^{\mu}$. By Kakutani’s fixed point theorem,
$C^{\mu}$ is non-empty. It follows that $\operatorname{Prob}(X)^{\mu}$ is non-
empty as well. ∎

Arguing as in the proof of Proposition 2.2, we deduce the following.

######  Proposition 2.6.

Let $C$ be a metrizable Choquet simplex and assume the locally compact second
countable group $G$ acts on $C$ continuously by affine transformations. Let
$\mu$ be an admissible probability measure on $G$ and consider its convolution
action on $C$. Then $C^{\mu}$ is a metrizable Choquet simplex.

###  2.3 Stiffness

We retain the setting of a locally compact second countable group $G$ acting
on a Banach space $E$ by linear isometries, along with the corresponding
continuous action of the semigroup $\operatorname{Prob}(G)$ on $E^{*}$ as
given in Equation (2.1).

######  Definition 2.7.

Let $C$ be a $\operatorname{Prob}(G)$-invariant subset of $E^{*}$. We say that
the action of $G$ on $C$ is _affinely $\mu$-stiff_ for a probability measure
$\mu\in\operatorname{Prob}(G)$, if any $\mu$-stationary point in $C$ is
$G$-invariant. A continuous action of $G$ on a Polish space $X$ is said to be
_topologically $\mu$-stiff_ if the corresponding action on
$\operatorname{Prob}(X)$ is affinely $\mu$-stiff.

When there is no risk of confusion, e.g when a Polish $G$-space $X$ does not
admit any a priori affine structure, we shall simply say _$\mu$ -stiff_. We
shall also often say that a Polish space $X$ is stiff as long as the action of
$G$ as well as the probability measure $\mu\in\operatorname{Prob}(G)$ are
implied by context.

We remark that in Definition 2.7 it could very well be the case that
$C^{\mu}$, the set of $\mu$-stationary points in $C$, is empty, in which case
the action is $\mu$-stiff in a vacuous sense. This is never the case when $C$
is weak-* compact due to the standard Kakutani argument. By Lemma 2.5, this is
also not the case for $\operatorname{Prob}(X)$, where
$X=\operatorname{Ext}(C)$ for some metrizable Choquet simplex $C$.

######  Lemma 2.8.

Let $\mu\in\operatorname{Prob}(G)$ admissible. The action of $G$ on a Polish
space $X$ is $\mu$-stiff if and only if any ergodic
$\nu\in\operatorname{Prob}(X)^{\mu}$ is invariant.

###### Proof.

This is an immediate application of Lemma 2.4 and the fact that
$\operatorname{Prob}(X)^{\mu}$ is a Polish simplex. ∎

We shall now state a few general and simple lemmas on stiffness.

######  Lemma 2.9.

Assume $G$ is a finite group and $\mu$ is an admissible probability measure on
$G$. Then every $G$-invariant convex subset $C\subset E^{*}$ is affinely
$\mu$-stiff.

###### Proof.

Observe that $\operatorname{Prob}(G)^{\mu}=\\{\lambda\\}$, where $\lambda$ is
the uniform measure on $G$. As a result $1/n\sum_{k=1}^{n}\mu^{k}$ converges
to $\lambda$ and therefore $C^{\mu}=C^{\lambda}=C^{G}$. ∎

######  Lemma 2.10.

Let $C$ be a convex $\operatorname{Prob}(G)$-invariant subset of $E^{*}$. Let
$\bar{G}$ be the quotient of $G$ by the kernel of $G\curvearrowright C$ and
let $\bar{\mu}$ be the pushforward of $\mu$ under the quotient map
$G\to\bar{G}$. Then $C$ is affinely $\mu$-stiff if and only if it is affinely
$\bar{\mu}$-stiff.

###### Proof.

Since the $G$-action on $C$ factorizes through $\bar{G}$, it is easy to see
that the corresponding $\operatorname{Prob}(G)$-action on $C$ factorizes
through $\operatorname{Prob}(\bar{G})$. This shows that
$C^{\mu}=C^{\bar{\mu}}$. As it is clear that $C^{G}=C^{\bar{G}}$, the
statement follows. ∎

######  Lemma 2.11.

Let $p:C_{1}\to C_{2}$ be a continuous affine surjective $G$-equivariant map
between two $\operatorname{Prob}(G)$-invariant convex sets and assume that the
fibers of $p$ are compact. If $C_{1}$ is affinely $\mu$-stiff then so is
$C_{2}$.

###### Proof.

For $c_{2}\in C_{2}^{\mu}$, the fiber $p^{-1}(c_{2})$ is non-empty, compact,
convex and closed under convolution by $\mu$. Thus we can choose a
$\mu$-stationary element $c_{1}\in p^{-1}(c_{2})$. Since $C_{1}$ is stiff,
$c_{1}$ must be $G$-invariant, and as a result, $c_{2}=p(c_{1})$ is
$G$-invariant. ∎

###  2.4 Relative stiffness

We introduce a relative version of stiffness for actions of a locally compact
second countable group $G$ endowed with an admissible probability measure
$\mu$.

######  Definition 2.12.

Let $p:X\to Y$ be a $G$-equivariant continuous map between Polish $G$-spaces.
$X$ is said to be _ $\mu$-stiff relative to $Y$_ if $X$ is measure preserving
relative to $Y$ with respect to any choice of an ergodic $\mu$-stationary
probability measure $\nu$ on $X$.

######  Lemma 2.13.

If $Y$ is stiff and $X$ is stiff relative to $Y$ then $X$ is stiff.
Conversely, if $X$ is stiff then it is stiff relative to $Y$ and, assuming
further that the fibers of $p$ are compact, $Y$ is stiff.

###### Proof.

Fix some $\nu\in\operatorname{Prob}(X)^{\mu}$ ergodic, and let
$\nu=\int\nu_{y}dp_{*}\nu$ be the disintegration of $\nu$ along $p$.

If $X$ is stiff relative to $Y$ and $Y$ is stiff then $p_{*}\nu$ is
$G$-invariant, and:

$g.\nu=\int_{Y}g.\nu_{y}dp_{*}\nu=\int_{Y}\nu_{g.y}dp_{*}\nu=\int_{Y}\nu_{y}dp_{*}\nu=\nu\,\,\,\,\,(\forall
g\in G)$

thus $\nu$ is $G$-invariant. This shows $X$ is stiff in light of Lemma 2.8.

The fact that if $X$ is stiff then it is stiff relative to $Y$ follows from
the uniqueness of the disintegration, as
$\nu=\int_{Y}g.\nu_{g^{-1}y}dp_{*}\nu$ implies $\nu_{y}=g.\nu_{g^{-1}y}$ a.e.
Further, if the fibers of $p$ are compact and $X$ is stiff then $Y$ is stiff
by Lemma 2.11. ∎

We introduce two cases in which relative stiffness is implied.

######  Proposition 2.14.

Let $p:X\to Y$ be a $G$-equivariant continuous map between Polish spaces and
assume all fibers are countable (or finite). Then $X$ is stiff relative to
$Y$.

This generalizes the well known fact that actions on countable spaces are
stiff.

###### Proof.

Let $\nu\in\operatorname{Prob}(X)^{\mu}$ ergodic and denote $\eta=p_{*}\nu$.
Disintegrate $\nu$ along $p$, namely write $\nu=\int_{Y}\nu_{y}\,{\rm d}\eta$
where $\nu_{y}\in\operatorname{Prob}(X)$ is defined for $\eta$-a.e $y$ and is
supported on the fiber $p^{-1}(y)$. For each $y\in Y$ for which $\nu_{y}$ is
defined, let $M_{y}$ be the (finite) set of all $x\in p^{-1}(y)$ that attain a
maximal $\nu_{y}$-measure (recall that $p^{-1}(y)$ is countable). Denote by
$M$ the union of all such $M_{y}$, and suppose first that we have managed to
show that $\nu(M)=1$. In that case, $\nu_{y}$ is the uniform measure supported
on $M_{y}$ for a.e $y$. The fact that $G$ preserves the measure class of $\nu$
implies that $M$ is essentially $G$-invariant. Thus writing $\nu$ according to
its integration formula we get that for all $g\in G$, $\nu_{y}(M_{y}\cap
gM_{g^{-1}y})=1$ for a.e $y\in Y$, or equivalently $g\nu_{g^{-1}y}=\nu_{y}$.
By Fubini, for a.e $y\in Y$, the equality $g\nu_{g^{-1}y}=\nu_{y}$ holds for
a.e $g\in G$. This equality must hold in fact for all $g\in G$ due to the
continuity of the action on the space of Lebesgue maps
$L(Y,\operatorname{Prob}(X))$ in the weak-$*$ topology induced from its
containment in $L^{1}(Y,C_{b}(X))^{*}$. Relative stiffness therefore follows.

In order to show that indeed $\nu(M)=1$, choose some Borel section $s:Y\to M$
of $p\\!\restriction_{M}$. The image $s(Y)$ is a Borel subset of $X$ (Suslin’s
theorem), and its $\nu$-measure is positive (as its $\nu_{y}$ measure is
positive for a.e $y\in Y$). For any $g\in G$ denote by $s^{g}:Y\to X$ the
section defined by $s^{g}(y)=gs(g^{-1}y)$. By the way $M$ was constructed, it
holds that $\nu_{y}\left(s(y)\right)\geq\nu_{y}\left(s^{g}(y)\right)$, where
equality holds if and only if $s^{g}(y)\in M$. In particular
$\nu(s(Y))\geq\nu(s^{g}(Y))$, and equality holds if and only if
$s^{g}(Y)\subseteq M$, up to null sets. The equality $\nu(s(Y))=\nu(s^{g}(Y))$
indeed holds for $\mu$-a.e $g\in G$ due to the stationarity of $\nu$:

$\nu(s(Y))=\int_{G}\nu\left(g.s(Y)\right)\,{\rm
d}\mu=\int_{G}\nu\left(s^{g}(Y)\right)\,{\rm d}\mu$

We see that for Haar-a.e $g\in G$, $s^{g}(Y)\subseteq M$ up to null sets,
because $\mu$ is admissible. This must hold for all $g\in G$ due to the
continuity of the $G$-action on $L^{\infty}(X,\nu)$ with the weak-* topology,
along with the observation that the set of all $f\in L^{\infty}(X,\nu)$ for
which $\chi_{M}\cdot f=f$ is closed.

Fix a countable dense subgroup $G_{0}\leq G$. The above shows that the set
$M_{0}:=\bigcup_{g\in G_{0}}s^{g}(Y)$ is contained in $M$, up to null sets.
But $M_{0}$ has positive measure as it contains $s(Y)$, and it is clearly
$G_{0}$-invariant. By the ergodicity of $\nu$ we conclude that $M_{0}$ has
full $\nu$-measure. In particular $M$ is of full measure, as desired. ∎

The second setting where we show relative stiffness is for extension by
compact groups:

######  Proposition 2.15.

Let $X$ be a Polish $G$-space and let $K$ be a compact group acting
continuously on $X$ and commuting with the $G$-action. Then $X$ is stiff
relatively to $X/K$.

The proof uses Zimmer’s notion of minimal cocycles which we now introduce (see
[Zim20] for further details). Let $\Omega$ be an ergodic Lebesgue $G$-space
and let $K$ be a second countable locally compact group. A _cocycle of $G$
over $\Omega$ taking values in $K$_ is a Lebesgue map $\alpha:G\times\Omega\to
K$ satisfying:

$\alpha(g_{1}g_{2},\omega)=\alpha(g_{1},g_{2}\omega)\alpha(g_{2},\omega);\,\,\,\,\,\left(g_{1},g_{2}\in
G,\,\omega\in\Omega\right)$

Given $\alpha$ one can construct the Lebesgue $G$-space
$\Omega\times_{\alpha}K$ defined as follows: as a Lebesgue space it is just
$\Omega\times K$ with the product $\sigma$-algebra and with the product of the
measure class of $\Omega$ with the Haar measure class on $K$. The $G$-action
on this space is defined by:
$g.\left(\omega,k\right)=(g.\omega,\alpha(g,\omega)k)$. It is not hard to see
that this turns $\Omega\times_{\alpha}K$ into a Lebesgue $G$-space. Two
cocycles $\alpha$ and $\beta$ are said to be _cohomologous_ if there exists an
a.e defined measurable map $\varphi:\Omega\to K$ for which:

$\beta(g,\omega)=\varphi(g.\omega)^{-1}\alpha(g,\omega)\varphi(\omega)$

for all $g\in G$ and for a.e $\omega\in\Omega$. In such a case
$\Phi:(\omega,k)\mapsto(\omega,\varphi(\omega)k)$ defines an isomorphism of
Lebesgue $G$-space between $\Omega\times_{\alpha}K$ and
$\Omega\times_{\beta}K$.

While $\alpha$ takes values in $K$, it could very well be that $\alpha$, or
perhaps some cocycle $\beta$ cohomologous to $\alpha$, takes values in a
proper subgroup of $K$. To that end define
$K_{\alpha}:=\overline{\left\langle\operatorname{Im}(\alpha)\right\rangle}$,
that is the minimal closed subgroup of $K$ containing
$\left\\{\alpha(g,\omega)\right\\}_{g\in G,\omega\in\Omega}$. $\alpha$ is
called a _minimal cocycle_ , if there is no $\beta$ cohomologous to $\alpha$
for which $K_{\beta}\subsetneq K_{\alpha}$.

######  Theorem 2.16 ([Zim20, Corollary 3.8]).

Let $\Omega$ be an ergodic Lebesgue $G$-space, and let $K$ be a second
countable _compact_ group. Then:

  1. 1.

Any cocycle $\alpha:G\times\Omega\to K$ is cohomologous to a minimal cocycle.

  2. 2.

If $\alpha$ and $\beta$ are cohomologous minimal cocycles then $K_{\alpha}$
and $K_{\beta}$ are conjugated in $K$.

  3. 3.

$\alpha$ is a minimal cocycle with $K=K_{\alpha}$ if and only if
$\Omega\times_{\alpha}K$ is ergodic.

###### Proof of Proposition 2.15.

Let $\nu\in\operatorname{Prob}(X)^{\mu}$ ergodic. We also assume as we may,
upon restriction, that $X=K\cdot\operatorname{supp}{\nu}$. We consider the
Borel map $X\to\operatorname{Sub}(K)$ taking each $x$ to its stabilizer
$K_{x}$ in $K$. Since the action of $G$ commutes with that of $K$, this map is
$G$-invariant, hence, as $\nu$ is ergodic, essentially constant on some
subgroup $K_{0}$. Moreover, this map is $K$-equivariant when $K$ acts on
$\operatorname{Sub}(K)$ by conjugation, so $K_{0}$ is normal in $K$. By
$X=K\cdot\operatorname{supp}{\nu}$ we get that $K_{0}$ acts trivially on $X$,
thus we assume as we may that $K_{0}$ is trivial. Replacing $X$ with a conull
subset which is $G\times K$ invariant, we assume as we may that $K$ acts
freely on $X$.

Denote $Y=X/K$ and let $p:X\to Y$ be the quotient map and endow $Y$ with the
measure $p_{*}\nu$. Using a choice of an a.e defined Borel section $s:Y\to X$
of $p$ we may identify $X\cong Y\times K$ as Borel spaces, and endow $X$ with
the product measure:

$\tilde{\nu}=p_{*}\nu\times\lambda\in\operatorname{Prob}(Y\times
K)\cong\operatorname{Prob}(X).$

where $\lambda$ is the Haar measure on $K$. This makes $X$ a Lebesgue
$G$-space. Define $\alpha:G\times Y\to K$ by setting $\alpha(g,y)$ to be the
unique element in $K$ for which $g.s(y)=\alpha(g,y).s(g.y)$. Then it is easy
to check that $\alpha$ is a cocycle and that $\rho:(y,k)\mapsto\sigma(y)k$
defines an isomorphism between the Lebesgue $G$-spaces $Y\times_{\alpha}K$ and
$X$. Using Theorem 2.16, we assume as we may that $\alpha$ is a minimal
cocycle, whose image is denoted by $K_{\alpha}\leq K$. We get a Borel,
$G$-invariant and $K$-equivariant map $X\to K/K_{\alpha}$. By
$\nu$-ergodicity, this map is essentially constant. Upon conjugating $\alpha$
by an element of $K$, we assume as we may that the essential image is the
basic coset $eK_{\alpha}$. It follows that $\nu$ is supported on the subset
$Y\times_{\alpha}K_{\alpha}$ of $Y\times_{\alpha}K=X$. We thus assume as we
may that $K=K_{\alpha}$ and $\alpha$ is a minimal cocycle.

Using Theorem 2.16 once again we see that $\tilde{\nu}$ is $G$-ergodic.
Noticing that $\tilde{\nu}$ is $\mu$-stationary, we get by Lemma 2.4 that it
is extremal as such. Observing that $\tilde{\nu}=\int_{K}k.\nu d\lambda(k)$,
we conclude that $\nu=\tilde{\nu}$. It follows that the disintegration map
$Y\to\operatorname{Prob}(Y\times_{\alpha}K)$ is given by
$y\mapsto\delta_{y}\times\lambda$, which is $G$-equivariant. Thus, indeed, $X$
is relatively measure preserving with respect to $Y=X/K$. ∎

###  2.5 Furstenberg-Poisson boundaries

The following result is the fundamental theorem of Furstenberg-Poisson theory,
see [BS06, §2] for a detailed overview. In fact, it could be taken as the
definition of Furstenberg-Poisson boundary.

######  Theorem 2.17 (Furstenberg).

Consider a locally compact second countable group $G$ and an admissible
probability measure $\mu$ on $G$. The Furstenberg-Poisson boundary of
$(G,\mu)$ is a Lebesgue $G$-space $\Pi(G,\mu)$ endowed with a $\mu$-stationary
probability measure $\nu$ which satisfies the following property: for every
metrizable compact convex $G$-set $C$, the correspondence that associates with
a Lebesgue $G$-map $\theta:\Pi(G,\mu)\to C$ the barycenter
$\operatorname{bar}{\theta_{*}\nu}\in C$ is a bijection between the collection
of all such Lebesgue $G$-maps and the space $C^{\mu}$ of stationary points in
$C$.

Moreover, every Lebesgue $G$-space $\Omega$ endowed with a stationary measure
$\eta$ which satisfies this property admits a unique morphism of Lebesgue
$G$-spaces $\Omega\to\Pi(G,\mu)$ and this morphism pushes $\eta$ to $\nu$. The
above uniquely defines $\Pi(G,\mu)$ up to a unique isomorphism of Lebesgue
$G$-spaces.

An immediate consequence of the above result is the following useful criterion
for invariance of a measure.

######  Lemma 2.18.

With the notation above, a point $c\in C^{\mu}$ is $G$-fixed iff the
corresponding map $\Pi(G,\mu)\to C$ is essentially constant.

We next discuss passage to finite index subgroups.

######  Definition 2.19.

Consider a locally compact second countable group $G$ and an open subgroup of
finite index $G^{\prime}<G$. Let $\mu$ be an admissible probability measure on
$G$. The associated _hitting time_ is the function

$\tau:G^{\mathbb{N}}\to\mathbb{N}\cup\infty,\quad(g_{1},g_{2},\ldots)\mapsto\inf\\{n\in\mathbb{N}\mid
g_{n}\cdot...\cdot g_{1}\in G^{\prime}\\}.$

By the law of large numbers, this function is $\mu^{\mathbb{N}}$-a.e finite,
thus the _hitting map_

$h:G^{\mathbb{N}}\to G^{\prime},\quad w=(g_{1},g_{2},\ldots)\mapsto
g_{\tau(w)}\cdot...\cdot g_{1}$

is $\mu^{\mathbb{N}}$-a.e defined. The _hitting measure_ on $G^{\prime}$ is
defined to be $\mu^{\prime}=h_{*}(\mu^{\mathbb{N}})$.

The hitting measure is discussed in a general context in [HLT14]. In
particular, the following is proved.

######  Proposition 2.20.

Consider a locally compact second countable group $G$ and an open subgroup of
finite index $G^{\prime}<G$. Let $\mu$ be an admissible probability measure on
$G$ and let $\mu^{\prime}$ be the corresponding hitting measure on
$G^{\prime}$. Then the restriction to $G^{\prime}$ of the Furstenberg-Poisson
boundary of $(G,\mu)$ is (uniquely isomorphic to) the Furstenberg-Poisson
boundary of $(G^{\prime},\mu^{\prime})$. Moreover, for every metrizable
compact convex $G$-set $C$, $C^{\mu}\subset C^{\mu^{\prime}}$.

###### Proof.

The first part of the proposition is proved in [HLT14, Corollary 2.14] under
the assumption that $G$ is discrete, however an examination of the proof
clearly reveals that the discreteness assumption is superfluous. The second
part of the proposition follows from the first by Theorem 2.17. ∎

######  Lemma 2.21.

Consider a locally compact second countable group $G$ and an open normal
subgroup of finite index $G^{\prime}\lhd G$. Let $\mu$ be an admissible
probability measure on $G$ and let $\mu^{\prime}$ be the corresponding hitting
measure on $G^{\prime}$. If a Polish $G$-space $X$ is $\mu^{\prime}$-stiff
then it is also $\mu$-stiff.

###### Proof.

Suppose $X$ is $\mu^{\prime}$-stiff, denote $C=\operatorname{Prob}(X)$ and let
$c\in C^{\mu}$. By Theorem 2.20 above, $c\in C^{\mu}\subseteq
C^{\mu^{\prime}}=C^{G^{\prime}}$. Let $\bar{\mu}$ be the pushforward measure
under the quotient map $G\to\bar{G}:=G/G^{\prime}$. The set $C^{G^{\prime}}$
is $G$-invariant, and the action of $G$ on it factorizes through $\bar{G}$. It
follows that by Lemma 2.10 that $c\in\left(C^{G^{\prime}}\right)^{\bar{\mu}}$.
But $\bar{G}$ is finite so by Lemma 2.9,
$c\in\left(C^{G^{\prime}}\right)^{\bar{G}}=C^{G}$. ∎

######  Lemma 2.22.

Consider a locally compact second countable group $G$ with an admissible
measure $\mu$ on it. Let $p:X\to Y$ be a $G$-equivariant continuous and
surjective map between Polish $G$-spaces. Assume that
$G^{\prime}:=\ker\left(G\curvearrowright Y\right)$ is an open subgroup of
finite index in $G$ and let $\mu^{\prime}$ denote the hitting measure on it.
If every fiber of $p$ is $\mu^{\prime}$-stiff then $X$ is $\mu$-stiff.

###### Proof.

By Lemma 2.21 it suffices to show that $X$ is $\mu^{\prime}$-stiff. Given
$\nu\in\operatorname{Prob}(X)$ we may disintegrate it along $p$, that is write
$\nu=\int_{Y}\nu_{y}\,{\rm d}\left(p_{*}\nu\right)$ where
$\nu_{y}\in\operatorname{Prob}(X)$ is defined for a.e $y\in Y$ and is
supported on the fiber $p^{-1}(y)$. Hence
$\mu^{\prime}*\nu=\int_{Y}\mu^{\prime}*\nu_{y}\,{\rm d}(p_{*}\nu)$ and each
$\mu^{\prime}*\nu_{y}$ is again a probability measure on $X$ which is
supported on $p^{-1}(y)$. Therefore if $\nu$ is $\mu^{\prime}$-stationary then
we have found two disintegration of $\nu$ along $p$, so by uniqueness of the
disintegration measures it must be that $\nu_{y}$ is $\mu^{\prime}$-stationary
for $p_{*}\nu$-a.e $y$. Since the action of $G^{\prime}$ on each fiber is
$\mu^{\prime}$-stiff, we see that the measures $\nu_{y}$ are
$G^{\prime}$-invariant a.e. Thus $\nu$ is $G^{\prime}$-invariant as desired. ∎

The following result of Margulis is a generalization of [Fur67, Theorem 3].

######  Proposition 2.23.

Let $G$ be a compactly generated locally compact group with left Haar measure
$\lambda$, and let $\Gamma<G$ be a lattice. Let $\psi:G\to[0,\infty)$ be a
continuous function with $\int_{G}\psi\,{\rm d}\lambda=1$, which is supported
on a compact subset that generates $G$ as a semigroup, and consider the
probability measure $\mu=\psi\cdot\lambda$ on $G$. Then there exists a fully
supported probability measure $\mu^{\prime}$ on $\Gamma$ such that the
restriction to $\Gamma$ of the Furstenberg-Poisson boundary of $(G,\mu)$ is
$\mu^{\prime}$-stationary and for every metrizable compact convex $G$-set $C$,
$C^{\mu}\subset C^{\mu^{\prime}}$. Moreover, given any function
$\phi:\Gamma\to(0,1]$, we may choose $\mu^{\prime}\leq\phi$.

###### Proof.

This is essentially the content of [Mar91, Proposition VI.4.1]. This reference
deals with an $S$-arithmetic setting, but this is superfluous and it serves as
a complete proof that the $\mu$-stationary measure on $\Pi(G,\mu)$ is
$\mu^{\prime}$-stationary for some fully supported
$\mu^{\prime}\in\operatorname{Prob}(\Gamma)$. Moreover, given any function
$\phi:\Gamma\to(0,1]$ one can impose throughout the construction the condition
$\mu^{\prime}(\gamma)\leq\phi(\gamma)$. The fact that $C^{\mu}\subset
C^{\mu^{\prime}}$ follows by Theorem 2.17. ∎

###  2.6 Furstenberg measures

We have defined the notion of a “Furstenberg measure” in the introduction, see
Definition 1.7. The following is a slight generalization of a classical result
of Furstenberg, [Fur67, Theorem 3], which regards the case in which $\Gamma$
is contained in the identity component of $\mathbf{G}(\mathbb{R})$.

######  Proposition 2.24.

Let $\mathbf{G}$ be an $\mathbb{R}$-algebraic group with a semisimple identity
component and let $\Gamma<\mathbf{G}(\mathbb{R})$ be a lattice. Then there
exists a Furstenberg measure $\mu$ on $\Gamma$.

###### Proof.

Denote $G=\mathbf{G}(\mathbb{R})$ and let $G^{o}$ be its identity component.
Let $P<G^{o}$ be a minimal parabolic subgroup and $K<G^{o}$ be a maximal
compact subgroup of this connected semisimple Lie group. Since all minimal
parabolic subgroups, as well as all maximal compact subgroups are conjugated
and $P$ and $K$ are self normalizers in $G^{o}$, we may identify the spaces
$G^{o}/P$ and $G^{o}/K$ with the corresponding sets of minimal parabolic and
maximal compact subgroups in $G^{o}$. Thus, the conjugation action of $G$ on
$G^{o}$ induces transitive actions on $G^{o}/P$ and $G^{o}/K$ correspondingly
and we may identify $G^{o}/P\simeq G/N_{G}(P)$ and $G^{o}/K\simeq G/N_{G}(K)$.

Endow $G$ with the left Haar measure $\lambda$, and let $\psi:G\to[0,\infty)$
be a continuous function with $\int_{G}\psi\,{\rm d}\lambda=1$ whose support
is compact and generates $G$ as a semigroup. and consider the probability
measure $\bar{\mu}=\psi\cdot\lambda$ on $G$. Find a $\bar{\mu}$-stationary
probability measure $\nu$ on the compact $G$-space $G/N_{G}(P)$. By
Proposition 2.23, there exists a fully supported probability measure $\mu$ on
$\Gamma$ such that $\nu$ is $\mu$-stationary. We choose $\bar{\mu}\leq\phi$
for $\phi:\Gamma\to(0,1]$ small enough so that $\mu$ has finite logarithmic
first moment and random walk entropy. We now use [Kai00, Theorem 10.7] to
deduce that $(G/N_{G}(P),\nu)$ is the Furstenberg-Poisson boundary of
$(\Gamma,\mu)$. Strictly speaking, it is assumed in this reference that $G$ is
connected, but since $G$ acts by isometries on the symmetric space $G^{o}/K$,
the use of the strip approximation [Kai00, Theorem 6.5] is intact. Since
$N_{G}(P)=\mathbf{N}(\mathbb{R})$ where $\mathbf{N}$ is the normalizer of the
Zariski closure of $P$, we get that
$G/N_{G}(P)=\mathbf{G}(\mathbb{R})/\mathbf{N}(\mathbb{R})$, as required to
show that $\mu$ is a Furstenberg measure on $\Gamma$. ∎

###### Proof of Proposition 1.8.

Let $\mathbf{G}$ be a $\mathbb{Q}$-algebraic group and $\Lambda$ an arithmetic
subgroup. Let $\mathbf{R}\lhd\mathbf{G}$ be the solvable radical of the
identity component subgroup in $\mathbf{G}$ and denote
$\bar{\mathbf{G}}=\mathbf{G}/\mathbf{R}$. Let $\bar{\Lambda}$ be the image of
$\Lambda$ in $\bar{\mathbf{G}}(\mathbb{Q})$. By [PRR93, Proposition 4.1]
$\bar{\Lambda}$ is an arithmetic subgroup of $\bar{\mathbf{G}}$, thus by a
classical theorem of Borel and Harish-Chandra it is a lattice in
$\bar{\mathbf{G}}(\mathbb{R})$. Using Proposition 2.24, we find a Furstenberg
measure $\bar{\mu}$ on $\bar{\Lambda}$. Then the Furstenberg-Poisson boundary
of $(\bar{\Lambda},\bar{\mu})$ is realized by
$\bar{\mathbf{G}}(\mathbb{R})/\bar{\mathbf{N}}(\mathbb{R})$, where
$\bar{\mathbf{N}}<\bar{\mathbf{G}}$ is the normalizer of some minimal
$\mathbb{R}$-parabolic subgroup $\bar{\mathbf{P}}$ of $\bar{\mathbf{G}}$. We
denote by $\mathbf{P}$ and $\mathbf{N}$ the preimages of $\bar{\mathbf{P}}$
and $\bar{\mathbf{N}}$ in $\mathbf{G}$ correspondingly and observe that
$\mathbf{P}$ is a minimal $\mathbb{R}$-parabolic subgroup and $\mathbf{N}$ is
its normalizer in $\mathbf{G}$. Since the kernel of $\Lambda\to\bar{\Lambda}$
is solvable, hence amenable, we can use [Kai02, Theorem 2] to find a
probability measure $\mu$ on $\Lambda$ such that the Furstenberg-Poisson
boundary of $(\Lambda,\mu)$ is the same as the Furstenberg-Poisson boundary of
$(\bar{\Lambda},\bar{\mu})$, that is
$\bar{\mathbf{G}}(\mathbb{R})/\bar{\mathbf{N}}(\mathbb{R})$. Identifying the
latter with $\mathbf{G}(\mathbb{R})/\mathbf{N}(\mathbb{R})$, we conclude that
$\mu$ is a Furstenberg measure on $\Lambda$. ∎

######  Remark 2.25.

We note that the same proof applies for S-arithmetic subgroups of
$k$-algebraic groups where $k$ is a number field and $S$ is a finite set of
places.

##  3 Characters

Throughout this section we fix a countable group $\Lambda$. All
representations are assumed to be unitary. For a comprehensive overview of the
theory of characters and more generally of dual spaces in harmonic analysis we
refer to [BdlH19].

###  3.1 Basics

A function $\varphi:\Lambda\to\mathbb{C}$ is called _positive-definite_ if for
any $n\in\mathbb{N}$, and for any group elements $g_{1},...,g_{n}\in\Lambda$,
the matrix $\left(\varphi(g_{i}^{-1}g_{j})\right)_{i,j}$ is positive semi-
definite, that is,
$\sum_{i,j}\bar{\alpha_{i}}\alpha_{j}\varphi(g_{i}^{-1}g_{j})\geq 0$ for any
choice of complex numbers $\alpha_{1},...,\alpha_{n}$. The space of such
functions is denoted by $\operatorname{PD}(\Lambda)$. It is not hard to see
that $\varphi(e)=\sup_{g\in\Lambda}|\varphi(g)|$ and in particular
$\operatorname{PD}(\Lambda)\subseteq l^{\infty}(\Lambda)$. We may therefore
speak of the weak-* topology on $\operatorname{PD}(\Lambda)$, which makes
$\operatorname{PD}(\Lambda)$ a close convex set. The space
$\operatorname{PD}_{1}(\Lambda)$ obtained by adding the normalization
condition $\varphi(e)=1$ is a then compact and convex. The weak-* topology on
$\operatorname{PD}_{1}(\Lambda)$ coincides with the topology of pointwise
convergence. The restriction to $\Lambda$ of any state on the (maximal)
$C^{*}$-algebra $C^{*}(\Lambda)$ is positive-definite and conversely any
$\varphi\in\operatorname{PD}_{1}(\Lambda)$ can be extended to
$C^{*}(\Lambda)$. This establishes a bijection between
$\operatorname{PD}_{1}(\Lambda)$ and the space of states on $C^{*}(\Lambda)$.

An element $\varphi\in\operatorname{PD}_{1}(\Lambda)$ which is conjugation
invariant is called a _trace_. Traces on $\Lambda$ correspond to tracial
states on $C^{*}(\Lambda)$. The collection of all traces
$\operatorname{Tr}(\Lambda)$ is again compact and convex, so it is natural to
speak of extreme points in $\operatorname{Tr}(\Lambda)$, that is traces that
cannot be written as a proper convex sum of two traces. Any extremal point of
this set is called a _character_ of $\Lambda$, and the collection of all
characters is denoted by $\operatorname{Ch}(\Lambda)$333Note that in some
texts, e.g [BBHP22], the term character refers to what we call here a trace..
Observe that $\operatorname{Tr}(\Lambda)$ is metrizable, as $\Lambda$ is
countable, thus its $G_{\delta}$-subset $\operatorname{Ch}(\Lambda)$ is Polish
(see [Phe01, Proposition 1.3]), but it is in general not compact, see e.g
Example 1.9. The space $\operatorname{Ch}(\Lambda)$ is sometimes referred to
as the _Thoma dual_ of $\Lambda$ as it serves as a dual space of $\Lambda$ for
harmonic analytic purposes. The following theorem may be thought of as a
Fourier transform on $\Lambda$.

######  Theorem 3.1 ([Tho64]).

$\operatorname{Tr}(\Lambda)$ is a Choquet simplex. That is, the barycenter map

$\mathcal{F}_{\Lambda}:\operatorname{Prob}(\operatorname{Ch}(\Lambda))\to\operatorname{Tr}(\Lambda),\quad\nu\mapsto\int_{\operatorname{Ch}(\Lambda)}\chi\,{\rm
d}\nu$

is a continuous affine bijection.

###### Proof of Proposition 1.6.

The Proposition follows from Lemma 2.5 in view of Theorem 3.1 and the
metrizability of $\operatorname{Tr}(\Lambda)$. ∎

###  3.2 The GNS construction

A _GNS triplet_ of $\Lambda$ is a triple $(\mathcal{H},\pi,\xi)$ where
$\mathcal{H}$ is a Hilbert space, $\pi:\Lambda\to\mathcal{U}(\mathcal{H})$ is
a representation, and $\xi$ is a cyclic vector. Given such, the function

$\displaystyle\varphi:g\mapsto\langle\pi(g)\xi,\xi\rangle,\;\;\;\;\;(g\in\Lambda)$
(3.1)

is in $\operatorname{PD}({\Lambda})$. Conversely, the GNS construction
associates to any $\varphi\in\operatorname{PD}(\Lambda)$ a GNS triplet
$(\mathcal{H},\pi,\xi)$ such that Eq. (3.1) holds, and this triplet is unique
up to an equivalence of representations that takes the respective cyclic
vectors to one another [BdlH19, Proposition 1.B.10]. Furthermore,
$\varphi\in\operatorname{PD}_{1}(\Lambda)$ if and only if $\xi$ is a unit
vector, and $\varphi$ is extremal if and only if $\pi$ is irreducible [BdlH19,
Proposition 1.B.14].

######  Lemma 3.2.

Let $\varphi\in\operatorname{PD}_{1}(\Lambda)$ with GNS triplet
$(\mathcal{H},\pi,\xi)$ and let $d\in\mathbb{N}$. If $\pi$ admits a
decomposition into $d$ irreducible subrepresentations then $\varphi$ is a
convex combination of $d$ extreme points of $\operatorname{PD}_{1}(\Lambda)$.

###### Proof.

Suppose $\pi=\bigoplus_{i=1}^{d}\pi_{i}$ is a decomposition into $d$
irreducible subrepresentations. For each $1\leq i\leq d$ consider the GNS
triplet $(\mathcal{H}_{i},\pi_{i},\xi_{i})$ where $\mathcal{H}_{i}$ is the
space of $\pi_{i}$, and $\xi_{i}$ is the projection of $\xi$ onto
$\mathcal{H}_{i}$. Denote the corresponding positive-definite functions to
these triplets by $\varphi_{i}$ and note that
$\varphi=\sum_{i=1}^{d}\varphi_{i}$. In particular, $\varphi$ is a convex
combination of the normalized positive-definite functions
$\frac{1}{\varphi_{i}(1)}\varphi_{i}$ (with respective coefficients
$\varphi_{i}(1)$). Since each $\pi_{i}$ are irreducible, each
$\frac{1}{\varphi_{i}(1)}\varphi_{i}$ is extremal. ∎

######  Definition 3.3.

Let $\varphi\in\operatorname{PD}_{1}(\Lambda)$. The _support_ of $\varphi$ is
$\operatorname{supp}(\varphi)=\Lambda\backslash\varphi^{-1}(0)$. The _kernel_
of $\varphi$ is $\ker\varphi:=\varphi^{-1}(1)$. $\varphi$ is called _faithful_
if $\ker\varphi=\\{e\\}$

######  Lemma 3.4.

For any $\varphi\in\operatorname{PD}_{1}(\Lambda)$, $\ker\varphi$ is a
subgroup of $\Lambda$, and $\varphi(k_{1}gk_{2})=\varphi(g)$ for all
$k_{1},k_{2}\in\ker\varphi$ and $g\in\Lambda$. In particular, if $\varphi$ is
a trace then $\ker\varphi$ is a normal subgroup and $\varphi$ descends to a
faithful trace of $\Lambda/\ker\varphi$.

###### Proof.

Let $\left(\mathcal{H},\pi,\xi\right)$ be a GNS triplet such that
$\varphi(g)=\left\langle\pi(g)\xi,\xi\right\rangle$. Then
$\ker\varphi=\left\\{g\in\Lambda\mid\pi(g)\xi=\xi\right\\}$ and in particular
it is a subgroup. Given $k_{1},k_{2}\in\ker\varphi$ we see that:

$\varphi(k_{1}gk_{2})=\left\langle\pi(k_{1}gk_{2})\xi,\xi\right\rangle=\left\langle\pi(g)\pi(k_{2})\xi,\pi(k_{1}^{-1})\xi\right\rangle=\left\langle\pi(g)\xi,\xi\right\rangle=\varphi(g)$

Now if $\varphi$ is a trace, $\ker\varphi$ is a normal subgroup, and by the
above it is constant on $\ker\varphi$-cosets. It therefore descends to a well
defined function $\bar{\varphi}$ of $\Lambda/\ker\varphi$. It is clear that
$\bar{\varphi}$ is a trace and that it is faithful. ∎

######  Definition 3.5.

Let $(\mathcal{H},\pi,\xi)$ be a GNS triplet of a subgroup
$\Gamma\leq\Lambda$. The _induced GNS triplet_ (from $\Gamma$ to $\Lambda$) is
$(\tilde{\mathcal{H}},\tilde{\pi},\tilde{\xi})$ where
$\tilde{\mathcal{H}}=l^{2}(\Lambda/\Gamma)\otimes\mathcal{H}$ is the
Hilbertian tensor product,
$\tilde{\pi}=\operatorname{Ind}_{\Gamma}^{\Lambda}\pi$ is the induced
representation, and $\tilde{\xi}=\delta_{e\Gamma}\otimes\xi$

Let $\Gamma$ be a subgroup of $\Lambda$. Given
$\varphi\in\operatorname{PD}_{1}(\Gamma)$ we define its _trivial extension_
$\tilde{\varphi}:\Lambda\to\mathbb{C}$ by:

$\tilde{\varphi}(g)=\begin{cases}\varphi(g)&g\in\Gamma\\\
0&g\notin\Gamma\end{cases}$

The fact that $\tilde{\varphi}$ is positive-definite follows by the following
Lemma:

######  Lemma 3.6.

Let $\Gamma\leq\Lambda$, and let $\varphi\in\operatorname{PD}_{1}(\Gamma)$
with GNS triplet $(\mathcal{H},\pi,\xi)$. Then the positive-definite function
corresponding to the induced GNS triplet
$(\tilde{\mathcal{H}},\tilde{\pi},\tilde{\xi})$ is $\tilde{\varphi}$.

###### Proof.

By definition of the induced representation, each $\tilde{\pi}(g)$ permutes
the pairwise orthogonal subspaces $\delta_{x\Gamma}\otimes\mathcal{H}$
($x\Gamma\in\Lambda/\Gamma$), and stabilizes the subspace
$\delta_{e\Gamma}\otimes\mathcal{H}$ if and only if $g\in\Gamma$. Moreover,
the subrepresentation of the restricted representation
$\operatorname{Res}_{\Gamma}^{\Lambda}\tilde{\pi}$ corresponding to the
$\Gamma$-invariant subspace $\delta_{e\Gamma}\otimes\mathcal{H}$ is simply
$\pi$. It follows that:

$\langle\tilde{\pi}(g)\tilde{\xi},\tilde{\xi}\rangle=\langle\operatorname{Res}^{\Lambda}_{\Gamma}\tilde{\pi}(g)\tilde{\xi},\tilde{\xi}\rangle=\langle\pi(g)\xi,\xi\rangle=\varphi(g)$

for $g\in\Gamma$, whereas
$\langle\tilde{\pi}(g)\tilde{\xi},\tilde{\xi}\rangle=0$ for $g\notin\Gamma$.
Altogether
$\tilde{\varphi}(g)=\langle\tilde{\pi}(g)\tilde{\xi},\tilde{\xi}\rangle$ for
all $g\in\Lambda$ ∎

######  Lemma 3.7.

Let $N$ be a normal subgroup of index $d<\infty$ in $\Lambda$, and let
$\varphi$ be an extreme point of $\operatorname{PD}_{1}(N)$. Then the trivial
extension $\tilde{\varphi}\in\operatorname{PD}_{1}({\Lambda})$ is a convex
combination of $d$ (or less) extreme points of
$\operatorname{PD}_{1}({\Lambda})$.

###### Proof.

Let $(\mathcal{H},\pi,\xi)$ be a GNS triplet associated with $\varphi$. Then
by Lemma 3.6, the induced GNS triplet
$(\tilde{\mathcal{H}},\tilde{\pi},\tilde{\xi})$ is a GNS triplet for
$\tilde{\varphi}$. As $N$ is normal
$\operatorname{Res}^{\Lambda}_{N}\operatorname{Ind}^{\Lambda}_{N}\pi=\bigoplus_{i=1}^{d}\pi^{t_{i}}$
where $t_{1},...,t_{d}$ are coset representatives for $N$ in $\Lambda$, and
$\pi^{t_{i}}$ is the representation of $N$ defined by
$n\mapsto\pi(t_{i}^{-1}nt_{i})$. Since $\varphi$ is extremal, $\pi$ is
irreducible and thus also each $\pi^{t_{i}}$. It follows that
$\operatorname{Ind}^{\Lambda}_{N}\pi$ is a direct sum of at most $d$
irreducible representations because this is true even when restricting to $N$.
Thus by Lemma 3.2 $\tilde{\varphi}$ is a convex combination of $d$ extreme
points of $\operatorname{PD}_{1}({\Lambda})$. ∎

###  3.3 Tracial representations

A _tracial von Neumann algebra_ is a pair $(M,\tau)$, where $M$ is a von
Neumann algebra and $\tau$ is a normal faithful trace on $M$. A _finite
factor_ is a tracial von Neumann algebra whose center consists only of scalar
multiples of the identity. A _tracial representation_ of a group $\Lambda$ is
a triple $(M,\tau,\pi)$ where $(M,\tau)$ is a tracial von Neumann algebra and
$\pi:\Lambda\to\mathcal{U}(\mathcal{M})$ is a representation of $\Lambda$ into
the group of unitary elements of $M$ such that $\pi(\Lambda)$ generates $M$ as
a von Neumann algebra. Two tracial representations are said to be equivalent
if there is an isomorphism between the two von Neumann algebras which
intertwines the traces and the representations. The following is a fundamental
result in the theory of characters.

######  Theorem 3.8 (Thoma).

There is a one to one correspondence between:

  1. 1.

Traces $\varphi:\Lambda\to\mathbb{C}$, and

  2. 2.

Isomorphism classes of tracial representations of $\Lambda$.

Moreover $\varphi$ is a character if and only if the corresponding von Neumann
algebra is a factor.

For the proof see [BdlH19, Lemmas 11.C.3 and 4], which are formulated in a
different, yet equivalent language, in view of [BdlH19, Proposition 6.B.5].
Briefly, given a tracial representation $(M,\tau,\pi)$, $\phi=\tau\circ\pi$ is
a trace on $\Lambda$ and we can retrieve $(M,\tau,\pi)$ from $\phi$ by the GNS
construction.

######  Corollary 3.9.

For an abelian group $\Lambda$, the space $\operatorname{Ch}(\Lambda)$
coincides with the Pontryagin dual $\hat{\Lambda}$.

###### Proof.

Given $\varphi\in\operatorname{Ch}(\Lambda)$ let $(M,\tau,\pi)$ be its
associated tracial representation. Then $M$ is a factor, and it is
commutative, thus isomorphic to $\mathbb{C}$. It follows that $\pi$ is a
homomorphism into the unit circle. ∎

######  Definition 3.10.

A von Neumann algebra is called _amenable_ if for some (equivalently, any)
embedding $M\subseteq\mathcal{B}(\mathcal{H})$, there exists a conditional
expectation $\mathcal{B}(\mathcal{H})\to M$. A trace
$\varphi\in\operatorname{Tr}(\Lambda)$, with corresponding tracial
representation $(M,\tau,\pi)$, is called _von Neumann amenable_ if $M$ is
amenable.

Any finite dimensional von Neumann algebra is amenable. It is well known that
all separable infinite-dimensional amenable finite factors are isomorphic to a
single von Neumann algebra $\mathcal{R}$, often referred to as the hyperfinite
$\operatorname{II_{1}}$-factor [MN36, Con76]. We refer to [AP17, Chapter 11]
for more on this manner.

###  3.4 Relative characters and stiffness

Traces and characters generalize to a relative setting. Let $\Gamma$ be a
group acting on $\Lambda$ by group automorphisms. Then $\Gamma$ acts on
$\operatorname{PD}_{1}(\Lambda)$ by $\varphi\mapsto\varphi^{\gamma}$, where
$\varphi^{\gamma}(g)=\varphi(\gamma^{-1}(g))$, and
$\operatorname{Tr}(\Lambda)$ is an invariant subset. We define the space of
_$\Gamma$ -traces_ of $\Lambda$, $\operatorname{Tr}_{\Gamma}(\Lambda)$, to be
the $\Gamma$-fixed points of $\operatorname{Tr}(\Lambda)$ and the space of
_$\Gamma$ -characters_ of $\Lambda$, $\operatorname{Ch}_{\Gamma}(\Lambda)$, to
be extreme points of $\operatorname{Tr}_{\Gamma}(\Lambda)$.

######  Theorem 3.11.

The compact convex space $\operatorname{Tr}_{\Gamma}(\Lambda)$ is a Choquet
simplex, that is

$\operatorname{Prob}(\operatorname{Ch}_{\Gamma}(\Lambda))\to\operatorname{Tr}_{\Gamma}(\Lambda)$

is a continuous affine bijection.

###### Proof.

By Theorem 3.1 $\operatorname{Tr}(\Lambda)$ is a Choquet simplex, thus by
Proposition 2.2 so is its subspace of $\Gamma$-fixed points. ∎

We now restrict to the case where $N$ is a normal subgroup of $\Lambda$ so
that $\Lambda$ acts on it by conjugation. For any
$\varphi\in\operatorname{Tr}(\Lambda)$ the restriction
$\varphi\\!\restriction_{N}$ is easily seen to be in
$\operatorname{Tr}_{\Lambda}(N)$. We call the mapping
$r:\varphi\to\varphi\\!\restriction_{N}$ the _restriction map_. Obviously it
is continuous, and it surjects onto $\operatorname{Tr}_{\Lambda}(N)$. Indeed,
the trivial extension $\tilde{\varphi}$ is a trace of $\Lambda$ whose
restriction to $N$ is $\varphi$.

######  Lemma 3.12.

The restriction map
$r:\operatorname{Tr}(\Lambda)\to\operatorname{Tr}_{\Lambda}(N)$ maps
$\operatorname{Ch}(\Lambda)$ surjectively onto
$\operatorname{Ch}_{\Lambda}(N)$. In particular if $N$ is in the center of
$\Lambda$, then $r$ maps $\operatorname{Ch}(\Lambda)$ onto the Pontryagin dual
$\hat{N}$ of $N$.

###### Proof.

The fact that $r$ takes $\operatorname{Ch}(\Lambda)$ to
$\operatorname{Ch}_{\Lambda}(N)$ is shown in [Tho64, Lemma 14]. The fact that
$r$ is surjective is shown in [Tho64, Lemma 16]. Finally, if $N$ is central
then $\Lambda$ acts trivially on $\operatorname{Tr}(N)$ and therefore
$\operatorname{Ch}_{\Lambda}(N)$ is equal to $\operatorname{Ch}(N)$ which is
simply $\hat{N}$ by Corollary 3.9. ∎

######  Proposition 3.13.

Suppose $N$ is a normal subgroup of finite index in $\Lambda$. Then each fiber
of the restriction map
$\operatorname{Ch}(\Lambda)\to\operatorname{Ch}_{\Lambda}(N)$ is of size at
most $[\Lambda:N]$.

The proof of Proposition 3.13 will be given in §3.5. The following is an
important conclusion regarding stiffness of actions on spaces of characters,
and will be used often.

######  Proposition 3.14.

Fix a group $\Gamma$ and an admissible measure $\mu$ on $\Gamma$. Assume that
$\Gamma$ acts by automorphisms on a group $\Lambda$ and let $N$ be a
$\Gamma$-invariant normal subgroup of $\Lambda$. Consider the following
statements regarding the corresponding $\Gamma$-actions on
$\operatorname{Ch}(\Lambda)$, $\operatorname{Ch}(N)$ and
$\operatorname{Ch}_{\Lambda}(N)$.

  1. 1.

$\Gamma\curvearrowright\operatorname{Ch}(N)$ is stiff.

  2. 2.

$\Gamma\curvearrowright\operatorname{Ch}_{\Lambda}(N)$ is stiff.

  3. 3.

$\Gamma\curvearrowright\operatorname{Ch}(\Lambda)$ is stiff.

Then 1 implies 2, 3 implies 2 if the fibers of
$\operatorname{Ch}(\Lambda)\to\operatorname{Ch}_{\Lambda}(N)$ are compact, and
all three are equivalent if $[\Lambda:N]<\infty$.

###### Proof.

By Theorem 3.11, $\operatorname{Tr}(N)$ and $\operatorname{Tr}_{\Lambda}(N)$
are Choquet simplices thus we have $\Gamma$-equivariant bijections

$\operatorname{Prob}(\operatorname{Ch}_{\Lambda}(N))\simeq\operatorname{Tr}_{\Lambda}(N)=\operatorname{Tr}(N)^{\Lambda}\simeq\operatorname{Prob}(\operatorname{Ch}(N))^{\Lambda}.$
(3.2)

Note, however that these are in general not homeomorphisms. Nevertheless, the
stiffness of all these spaces are equivalent by Lemma 2.5.

Stiffness of the Polish space $\operatorname{Ch}(N)$ is by definition
equivalent to stiffness of the convex space
$\operatorname{Prob}(\operatorname{Ch}(N))$ which implies the stiffness of its
$\Gamma$-invariant convex subspace
$\operatorname{Prob}(\operatorname{Ch}(N))^{\Lambda}$, hence by Equation
(3.2), of $\operatorname{Prob}(\operatorname{Ch}_{\Lambda}(N))$ which is
equivalent (by definition) to the stiffness of the Polish space
$\operatorname{Ch}_{\Lambda}(N)$. This shows that $1$ implies $2$.

By Lemma 3.12, the map
$\operatorname{Ch}(\Lambda)\to\operatorname{Ch}_{\Lambda}(N)$ is surjective,
thus by Lemma 2.11, if its fibers are compact then $3$ implies $2$.

Assume that $[\Lambda:N]<\infty$. Then $2$ implies $3$ by Proposition 3.13 and
Proposition 2.14. Finally to see that $2$ implies $1$ note that Equation (3.2)
identifies $\operatorname{Ch}_{\Lambda}(N)$ with ergodic $\Lambda$-invariant
probability measures on $\operatorname{Ch}(N)$. All such measures are uniform
measures on finite orbits thus $\operatorname{Ch}_{\Lambda}(N)$ is identified
with the space of orbits $\operatorname{Ch}(N)/\Lambda$. The statement then
follows from Proposition 2.14. ∎

######  Corollary 3.15.

Fix a group $\Gamma$ and an admissible measure $\mu$ on $\Gamma$. Assume that
$\Gamma$ acts by automorphisms on a group $\Lambda$. If $\Lambda$ is finitely
generated and $\Lambda_{0}<\Lambda$ is a $\Gamma$-invariant subgroup of finite
index then $\Gamma\curvearrowright\operatorname{Ch}(\Lambda)$ is stiff iff
$\Gamma\curvearrowright\operatorname{Ch}(\Lambda_{0})$ is stiff. If, further,
$\Lambda$ is residually finite and $F\lhd\Lambda$ is a $\Gamma$-invariant
finite normal subgroup then $\Gamma\curvearrowright\operatorname{Ch}(\Lambda)$
is stiff iff $\Gamma\curvearrowright\operatorname{Ch}(\Lambda/F)$ is stiff.

###### Proof.

Assume $\Lambda$ is finitely generated and let $\Lambda_{0}<\Lambda$ be a
$\Gamma$-invariant subgroup. Using the fact that $\Lambda$ is finitely
generated, it has only finitely many subgroups of index
$[\Lambda:\Lambda_{0}]$, thus their intersection $N$ is a characteristic
normal subgroup of finite index in $\Lambda$ which is contained in
$\Lambda_{0}$. In particular, $N$ is $\Gamma$-invariant. By $1\Leftrightarrow
3$ in Proposition 3.14 we get that
$\Gamma\curvearrowright\operatorname{Ch}(\Lambda)$ is stiff iff
$\Gamma\curvearrowright\operatorname{Ch}(N)$ is stiff iff
$\Gamma\curvearrowright\operatorname{Ch}(\Lambda_{0})$ is stiff.

Assume further that $\Lambda$ is residually finite and let $F\lhd\Lambda$ be a
$\Gamma$-invariant finite normal subgroup. Using the fact that $\Lambda$ is
residually finite, we find a normal subgroup of finite index $N\lhd\Lambda$
such that $F$ injects into $\Lambda/N$. Arguing as above, we assume as we may
that $N$ is characteristic, thus $\Gamma$-invariant. Therefore, the
$\Gamma$-action descents to an action on the finite group $\Lambda/N$. We
consider the $\Gamma$-equivariant injection
$\Lambda\to\Lambda/F\times\Lambda/N$ and identify $\Lambda$ with its image,
which is of finite index. Applying twice the first part of the corollary, we
get that $\Gamma\curvearrowright\operatorname{Ch}(\Lambda)$ is stiff iff
$\Gamma\curvearrowright\operatorname{Ch}(\Lambda/F\times\Lambda/N)$ is stiff
iff $\Gamma\curvearrowright\operatorname{Ch}(\Lambda/F\times\\{e\\})$ is
stiff. We conclude that $\Gamma\curvearrowright\operatorname{Ch}(\Lambda)$ is
stiff iff $\Gamma\curvearrowright\operatorname{Ch}(\Lambda/F)$ is stiff. ∎

###  3.5 Relative traces as positive-definite functions with kernels

If $\varphi\in\operatorname{Tr}(\Lambda)$ is an extreme point of
$\operatorname{PD}_{1}({\Lambda})$, then it is also an extreme point of
$\operatorname{Tr}({\Lambda})$, but the converse does not hold in general. To
this matter we introduce a perspective used to bridge over this gap, and in
turn to complete the proof of Proposition 3.13. We start with the following
general statement:

######  Proposition 3.16.

Consider a group $\Gamma$ acting on $\Lambda$ by group automorphisms and the
associated semidirect product $\Gamma\ltimes\Lambda$. Denote by
$p:\Gamma\ltimes\Lambda\to\Lambda$ the (non-homomorphic) map
$(\gamma,g)\mapsto g$. Then $\varphi\mapsto\varphi\circ p$ defines an
injective affine continuous map
$\operatorname{PD}_{1}({\Lambda})^{\Gamma}\to\operatorname{PD}_{1}({\Gamma\ltimes\Lambda})$.
The image is the set of all
$\psi\in\operatorname{PD}_{1}({\Gamma\ltimes\Lambda})$ with
$\Gamma\subseteq\ker\psi$, and in particular it is a face of
$\operatorname{PD}_{1}({\Gamma\ltimes\Lambda})$.

###### Proof.

Let $\varphi\in\operatorname{PD}_{1}({\Lambda})^{\Gamma}$. Given
$n\in\mathbb{N}$, $\gamma_{1},...,\gamma_{n}\in\Gamma$,
$g_{1},...,g_{n}\in\Lambda$ and $\alpha_{1},...,\alpha_{n}\in\mathbb{C}$
write:

$\displaystyle\sum_{i,j}\overline{\alpha_{i}}\alpha_{j}\left(\varphi\circ
p\right)\left((\gamma_{i},g_{i})^{-1}\cdot(\gamma_{j},g_{j})\right)=\sum_{i,j}\overline{\alpha_{i}}\alpha_{j}\varphi\left(\gamma_{i}^{-1}\left(g_{i}^{-1}g_{j}\right)\right)=\sum_{i,j}\overline{\alpha_{i}}\alpha_{j}\varphi\left(g_{i}^{-1}g_{j}\right)\geq
0$

Hence $\varphi\circ p\in\operatorname{PD}_{1}(\Gamma\ltimes\Lambda)$ (a more
conceptual argument is obtained via the associated GNS representation). The
fact that the mapping $\varphi\mapsto\varphi\circ p$ is injective, affine and
continuous is straightforward.

Let $F$ denote the set of all
$\psi\in\operatorname{PD}_{1}({\Gamma\ltimes\Lambda})$ with
$\Gamma\subseteq\ker\psi$. As $\psi$ attains values in the unit disk in
$\mathbb{C}$, it follows that $F$ is a face of
$\operatorname{PD}_{1}({\Gamma\ltimes\Lambda})$. Moreover, $\varphi\circ p\in
F$ for any $\varphi\in\operatorname{PD}_{1}(\Lambda)^{\Gamma}$, so it is left
to show surjectivity onto $F$. If
$\psi\in\operatorname{PD}_{1}(\Gamma\ltimes\Lambda)$ then clearly
$\psi\\!\restriction_{\Lambda}\in\operatorname{PD}_{1}({\Lambda})$. Assuming
moreover that $\psi\in F$ we deduce by Lemma 3.4 that for all
$\gamma\in\Gamma$ and $g\in\Lambda$:

$\psi\\!\restriction_{\Lambda}(\gamma(g))=\psi\left(e_{\Gamma},\gamma(g)\right)=\psi\left((\gamma,e_{\Lambda})(e_{\Gamma},g)(\gamma^{-1},e_{\Lambda})\right)=\psi(e_{\Gamma},g)=\psi\\!\restriction_{\Lambda}(g)$

showing that
$\psi\\!\restriction_{\Lambda}\in\operatorname{PD}_{1}(\Lambda)^{\Gamma}$, and

$\psi(\gamma,g)=\psi((e_{\Gamma},g)(\gamma,e_{\Lambda}))=\psi(e_{\Gamma},g)=\psi\\!\restriction_{\Lambda}\circ
p(\gamma,g)$

showing that $\psi=\psi\\!\restriction_{\Lambda}\circ p$. ∎

######  Example 3.17.

Fix a group $\Lambda$ and let $\Gamma=\Lambda$ act on it by inner
automorphisms. We get the identification
$\operatorname{PD}_{1}({\Lambda})^{\Gamma}=\operatorname{Tr}({\Lambda})$. Note
that we may also identify in this case
$\Gamma\ltimes\Lambda\simeq\Lambda\times\Lambda$, by
$(\gamma,\lambda)\mapsto(\gamma,\lambda\gamma)$. Under these identifications,
the map
$\operatorname{PD}_{1}({\Lambda})^{\Gamma}\to\operatorname{PD}_{1}({\Gamma\ltimes\Lambda})$
considered in Proposition 3.16, identifies $\operatorname{Tr}({\Lambda})$ with
the subset of elements of $\operatorname{PD}_{1}(\Lambda\times\Lambda)$ which
are trivial on the diagonal subgroup. This is a well known identification.

The following is a generalization of the previous example.

######  Example 3.18.

Consider a group $\Gamma$ acting on $\Lambda$ by group automorphisms and
assume that the image of $\Gamma$ in $\operatorname{Aut}(\Lambda)$ contains
all inner automorphisms. Then
$\operatorname{PD}_{1}({\Lambda})^{\Gamma}=\operatorname{Tr}_{\Gamma}(\Lambda)$,
so that $\operatorname{Tr}_{\Gamma}(\Lambda)$ embeds as a face of
$\operatorname{PD}_{1}(\Gamma\ltimes\Lambda)$. In particular
$\varphi\in\operatorname{Tr}_{\Gamma}(\Lambda)$ belongs to
$\operatorname{Ch}_{\Gamma}(\Lambda)$ if and only if $\varphi\circ p$ is an
extreme point of $\operatorname{PD}_{1}(\Gamma\ltimes\Lambda)$.

We will use the last example in the course of the proof of the following
lemma.

######  Lemma 3.19.

Let $N$ be a normal subgroup of index $d<\infty$ in $\Lambda$, and let
$\varphi\in\operatorname{Ch}_{\Lambda}(N)$. Then the trivial extension
$\tilde{\varphi}\in\operatorname{Tr}({\Lambda})$ decomposes as a convex
combination of $d$ (or less) characters of $\Lambda$.

###### Proof.

Consider the inner action of $\Lambda$ on itself, as well as on the invariant
subset $N$. With respect to these actions form the semidirect products
$\Lambda\ltimes\Lambda$ and $\Lambda\ltimes N$ and let
$p_{\Lambda}:\Lambda\ltimes\Lambda\to\Lambda$ and $p_{N}:\Lambda\ltimes N\to
N$ denote the (set-theoretic) projections onto the second coordinates. By
Proposition 3.16, showing that $\tilde{\varphi}$ is a convex combination of
$d$ characters is the same as showing that $\tilde{\varphi}\circ p_{\Lambda}$
is a convex combination of $d$ extreme points of
$\operatorname{PD}_{1}({\Lambda\ltimes\Lambda})$.

Now $\varphi$ is an extreme point of $\operatorname{Tr}_{\Lambda}(N)$ and
therefore by Proposition 3.16 $\varphi\circ p_{N}$ is an extreme point of
$\operatorname{PD}_{1}({\Lambda\ltimes N})$. Hence, by Lemma 3.7
$\widetilde{\varphi\circ p_{N}}$, the trivial extension of $\varphi\circ
p_{N}$ from $\Lambda\ltimes N$ to $\Lambda\ltimes\Lambda$, is a convex
combination of $d$ extreme points of
$\operatorname{PD}_{1}({\Lambda\ltimes\Lambda})$. But clearly
$\tilde{\varphi}\circ p_{\Lambda}=\widetilde{\varphi\circ p_{N}}$, which
completes the proof. ∎

###### Proof of Proposition 3.13.

Let $\varphi\in\operatorname{Ch}_{\Lambda}(N)$ and let $\tilde{\varphi}$ be
its trivial extension. By Lemma 3.19, $\tilde{\varphi}$ is a convex
combination of $[\Lambda:N]$ characters of $\Lambda$. But according to [Tho64,
Lemma 15], each $\psi\in\operatorname{Ch}(\Lambda)$ with
$\psi\\!\restriction_{N}=\varphi$ appears as one of those $[\Lambda:N]$ many
characters in the decomposition of $\tilde{\varphi}$. It follows that the
fiber of $\varphi$ under the restriction map
$\operatorname{Ch}(\Lambda)\to\operatorname{Ch}_{\Lambda}(N)$ is of size at
most $[\Lambda:N]$. ∎

##  4 Stiffness of arithmetic groups

The goal of this section is to prove Theorem 4.1 bellow and to deduce from it
Theorems C and D.

######  Theorem 4.1.

Let $U$ be a connected, simply connected nilpotent real Lie group and $H\leq
U$ a lattice. Let $S$ be a reductive group with compact center acting
continuously by automorphisms on $U$ and let $\Gamma<S$ be a lattice
preserving $H$. Let $\mu$ be a Furstenberg measure on $\Gamma$. Then the
action of $\Gamma$ on $\operatorname{Ch}(H)$ is $\mu$-stiff.

Due to Lemma 2.21, there is no loss of generality in assuming $S$ to be
connected. We denote by $S_{\mathrm{nc}}$ the subgroup of $S$ generated by
unipotents. This is a semisimple group with no compact simple factors. We
denote by $S_{\mathrm{c}}$ the maximal compact normal subgroup of $S$. We
obtain a finite central extension $S_{\mathrm{nc}}\times S_{\mathrm{c}}\to S$.
Upon replacing $S$ with this central extension, replacing $\Gamma$ with its
preimage and replacing $\mu$ with an arbitrary lift, we assume as we may that
$S=S_{\mathrm{nc}}\times S_{\mathrm{c}}$. Note that indeed, $\mu$ is still a
Furstenberg measure on $\Gamma$.

We note that $S$ has a canonical real algebraic structure and every continuous
linear representation of it is automatically real algebraic. By the Borel
Density Theorem, the image of $\Gamma$ is Zariski dense in $S_{\mathrm{nc}}$.
Upon replacing $S_{\mathrm{c}}$ with the closure of the image of $\Gamma$, we
assume as we may that this image is dense. We thus have that $\Gamma$ is
Zariski dense in $S$.

We let $P<S$ be a minimal parabolic and note that $P=P_{\mathrm{nc}}\times
S_{\mathrm{c}}$, where $P_{\mathrm{nc}}$ is a minimal parabolic in
$S_{\mathrm{nc}}$. Thus $S/P\simeq S_{\mathrm{nc}}/P_{\mathrm{nc}}$. Endowing
this space with the Haar measure class, it is $\Gamma$-isomorphic as a
Lebesgue space to $\Pi(\Gamma,\mu)$, by the assumption that $\mu$ is a
Furstenberg measure.

###  4.1 The abelian case

We start by proving the following special case.

######  Proposition 4.2 (cf. [Fur98]).

Theorem 4.1 holds when $U$ is abelian.

This formulation is merely a slight generalization of Furstenberg’s original
result, yet this generalization is essential to our discussion, so we give
here its full proof. We will need the following definition.

######  Definition 4.3.

Let $k$ be a field and let $\mathbf{L}$ be a $k$-algebraic group. An algebraic
subgroup $\mathbf{H}$ defined over $k$ is said to be _epimorphic_ in
$\mathbf{L}$ if for any $k$-representation $V$ of $\mathbf{L}$, a vector $v\in
V$ is fixed by $\mathbf{L}$ as soon as it is fixed by $\mathbf{H}$.

Any $k$-parabolic subgroup $\mathbf{P}$ of a connected $k$-algebraic group
$\mathbf{L}$ is epimorphic. To see that, suppose
$\rho:\mathbf{L}\to\mathbf{\operatorname{GL}}(V)$ is a representation and that
$v\in V$ is $\mathbf{P}$-fixed. Thus the orbit map $\mathbf{L}\to V$ given by
$g\mapsto\rho(g)v$ factorizes to a map $\mathbf{L}/\mathbf{P}\to V$. But this
map must be constant as $\mathbf{L}/\mathbf{P}$ is a projective variety,
meaning that $v$ is $\mathbf{L}$-fixed.

The proof of Proposition 4.2 is based on the following measure rigidity
theorem due to Mozes.

######  Theorem 4.4 ([Moz95, Theorem 2]).

Let $\mathbf{G}$ be an $\mathbb{R}$-algebraic group, let
$\mathbf{L}\leq\mathbf{G}$ be an $\mathbb{R}$-algebraic subgroup generated by
connected unipotent $\mathbb{R}$-subgroups and let $\mathbf{H}<\mathbf{L}$ be
a connected epimorphic $\mathbb{R}$-algebraic subgroup subgroup of
$\mathbf{L}$. Consider a discrete subgroup $\Gamma$ of
$\boldsymbol{\mathbf{G}}(\mathbb{R})$. Then any probability measure on
$\mathbf{G}(\mathbb{R})/\Gamma$ is $\mathbf{L}(\mathbb{R})$-invariant as soon
as it is $\mathbf{H}(\mathbb{R})$-invariant.

###### Proof of Proposition 4.2.

In this case $U\cong\mathbb{R}^{m}$ and $\operatorname{Ch}(H)\cong U/H$ is a
torus which we denote by $X$. Let $\nu\in\operatorname{Prob}(X)^{\mu}$ and
consider the corresponding boundary map given by Theorem 2.17,

$\Theta:S/P\simeq
S_{\mathrm{nc}}/P_{\mathrm{nc}}\simeq\Pi(\Gamma,\mu)\to\operatorname{Prob}(X).$

We will show that $\Theta$ is essentially constant, thus $\nu$ is
$\Gamma$-invariant (Lemma 2.18).

We let $\,{\rm d}s$ be the Haar measure on $S$, normalized so that it induces
a probability measure on $S/\Gamma$. We consider the quotient map $q:S\to
S/P\simeq S_{\mathrm{nc}}/P_{\mathrm{nc}}$ and set $\theta:=\Theta\circ q$.
This is an a.e defined $\Gamma$-equivariant measurable map.

We consider the space $S\times_{\Gamma}X$, defined as $S\times X$ quotiented
by the $\Gamma$-action $\gamma.\left(s,x\right)=\left(s\gamma^{-1},\gamma
x\right)$, and we let $S$ act on this space by left multiplication on the
first coordinate. We define the measure $\tilde{\sigma}$ on $S\times X$ by:

$\tilde{\sigma}=\int_{S}\delta_{s}\times\theta_{s^{-1}}\,{\rm d}s.$

It is easy to see that $\tilde{\sigma}$ is both $\Gamma$-invariant, for the
action considered above, and $P$-invariant, for the restriction of the left
$S$-action. Hence $\tilde{\sigma}$ descends to a $P$-invariant probability
measure $\sigma$ on $S\times_{\Gamma}X$.

We define $G=S\ltimes U$. By the fact that every linear representation of $S$
is real algebraic, we have that $G$ is a real algebraic group. We set
$\Lambda=\Gamma\ltimes H$ and notice that this is a lattice in $G$. In fact,
we may identify

$G/\Lambda=(S\ltimes U)/\Lambda\simeq((S\ltimes U)/H)/\Gamma\simeq(S\times
X)/\Gamma=S\times_{\Gamma}X.$

Using this identification, we endow $G/\Lambda$ with the measure $\sigma$
constructed above.

We are now in the setting of theorem 4.4: $G$ is a real algebraic group,
$\Lambda\leq G$ is discrete, $S_{\mathrm{nc}}$ is a connected real algebraic
subgroup of $G$ generated by unipotents, and $P_{\mathrm{nc}}$ is an
epimorphic subgroup of $S_{\mathrm{nc}}$ that fixes $\sigma$. We conclude that
$\sigma$ is $S_{\mathrm{nc}}$-invariant. In turn, we get that $\tilde{\sigma}$
is $S_{\mathrm{nc}}$-invariant. Since for $t\in S_{\mathrm{nc}}$ we have

$t.\tilde{\sigma}=\int_{S}\delta_{ts}\times\theta_{s^{-1}}\,{\rm
d}s=\int_{S}\delta_{s}\times\theta_{s^{-1}t}\,{\rm d}s.$

The $S_{\mathrm{nc}}$-invariance of $\tilde{\sigma}$ implies that
$\theta_{s^{-1}t}=\theta_{s^{-1}}$ almost surely, meaning that $\theta$, and
therefore also $\Theta$, is $S_{\mathrm{nc}}$-invariant. It follows that
$\Theta:S_{\mathrm{nc}}/P_{\mathrm{nc}}\to\operatorname{Prob}(X)$ is
essentially constant, as desired. ∎

###  4.2 Proof of Theorem 4.1

The path from the abelian case to the nilpotent case is based on central
induction of characters.

######  Definition 4.5.

The _FC-center_ of a group $H$ is the union of all of its finite conjugacy
classes, and is denoted by $\operatorname{FC}(H)$. $H$ is called _centrally
inductive_ if every character $\varphi$, viewed as a faithful characters of
$H/\ker\varphi$, is supported on
$\operatorname{FC}\left(H/\ker\varphi\right)$.

The FC-center is a characteristic subgroup that plays an important role in the
character theory of countable groups (see for example the recent result in
[Bek21]), and in greater extent for nilpotent groups:

######  Theorem 4.6 ([How77, CM84]).

Any finitely generated nilpotent group is centrally inductive.

We will use the following simple Lemma:

######  Lemma 4.7.

Let $H$ be a nilpotent group. Then any non-trivial normal subgroup intersects
the center non-trivially.

###### Proof.

Let $Z=Z_{1}\leq...\leq Z_{m}=H$ be the central ascending sequence of $H$ and
assume by contradiction that $N\lhd H$ is a non-trivial normal subgroup
intersecting $Z$ trivially. Choose $x\in N$ which belongs to $Z_{i}$ for a
minimal $i$. Since $x\notin Z$, we can find $y\in H$ with $[x,y]\in N$ non-
trivial, contradicting the minimality of $i$. ∎

The proof of Theorem 4.1 is by induction on the dimension of $U$. Let
$\nu\in\operatorname{Prob}(\operatorname{Ch}(H))^{\mu}$ ergodic. Consider the
map $k:\operatorname{Ch}(H)\to\operatorname{Sub}(Z)$ that takes each character
to the intersection of its kernel with the center $Z$ of $H$. Then $k_{*}\nu$
is an ergodic $\mu$-stationary probability measure on $\operatorname{Sub}(Z)$.
But $\operatorname{Sub}(Z)$ is countable and therefore by Lemma 2.14 (applied
for $X=\operatorname{Sub}(Z)$ and $Y$ being a singleton) $k_{*}\nu$ must be
$\Gamma$-invariant, thus by ergodicity, a uniform measure on some finite orbit
$\mathcal{D}\subseteq\operatorname{Sub}(Z)$.

#### Case a: $\mathcal{D}$ contains a non-trivial subgroup.

Fix a non-trivial subgroup $K\in\mathcal{D}$. By the assumptions on $U$, the
exponential map $\exp:\operatorname{Lie}(U)\to U$ is bijective. Let
$\tilde{K}$ denote the Lie subgroup of $U$ corresponding to the
$\mathbb{R}$-span of $\log(K)$. The group $Z$ is finitely generated free
abelian [Rag72, Theorem 2.18] so that $K$ is infinite. It follows that
$\tilde{K}$ is a Lie subgroup of the center of $U$ of positive dimension.
Moreover $K$ is a finite index subgroup of $H\cap\tilde{K}$ and is a lattice
in $\tilde{K}$.

Let $\Gamma^{\prime}$ be the kernel of the action of $\Gamma$ on
$\mathcal{D}$, endowed with the hitting measure $\mu^{\prime}$. By Proposition
2.20 $\mu^{\prime}$ is a Furstenberg measure. As $K$ is
$\Gamma^{\prime}$-invariant so is $\tilde{K}$ [Rag72, Theorem 2.11]. By the
Zariski density of $\Gamma^{\prime}$ in $S$, $\tilde{K}$ is also $S$-invariant
and therefore $S$ acts on $U_{0}:=U/\tilde{K}$ by continuous automorphisms.
Now $H_{0}:=H/\left(H\cap\tilde{K}\right)$ is discrete in $U_{0}$ and moreover
a lattice [Rag72, Theorem 2.1]. By induction hypothesis,
$\operatorname{Ch}(H_{0})$ is $\mu^{\prime}$-stiff. Since this holds for each
$K\in\mathcal{D}$, me may apply Proposition 2.22 to the map
$k:\operatorname{supp}(\nu)\to\mathcal{D}$ to conclude that the action of
$\Gamma$ on $\operatorname{supp}(\nu)$ is $\mu$-stiff. It follows that $\nu$
is $\Gamma$-invariant.

#### Case b: $\mathcal{D}$ consists only of the trivial subgroup.

By Lemma 4.7 we get in this case that $\nu$-a.e character of $H$ is faithful.
Since $H$ is a lattice in $U$, it is nilpotent, finitely generated and
torsion-free [Rag72, Theorem 2.18]. Therefore it is centrally inductive, and
its FC-center is its center, $Z$ (see [Rob72, Theorem 4.37]). Thus $\nu$-a.e
$\chi\in\operatorname{Ch}(H)$ is supported on $Z$. Writing
$\varphi=\mathcal{F}_{H}(\nu)$ for the corresponding trace (Theorem 3.1), we
get that $\varphi$ is supported on $Z$:

$\displaystyle\forall h\in H\backslash
Z:\,\,\varphi(h)=\int_{\operatorname{Ch}(H)}\chi(h)d\nu(\chi)=0$ (4.1)

The pushforward measure $r_{*}\nu$ under the $\Gamma$-equivariant restriction
map $r:\operatorname{Ch}(H)\to\operatorname{Ch}(Z)\simeq\hat{Z}$ is a
$\mu$-stationary probability measure. $Z$ is a lattice in the center of $U$
[Rag72, Proposition 2.17]. Since $Z$ is $\Gamma$-invariant and the center of
$U$ is $S$-invariant, we get by Proposition 4.2 that $r_{*}\nu$ is
$\Gamma$-invariant. The trace corresponding to $r_{*}\nu$ via the Fourier
transform is $\varphi\\!\restriction_{Z}$ and so we see that

$\displaystyle\forall\gamma\in\Gamma,\;\;h\in
Z:\varphi^{\gamma}(h)=\varphi(h)$ (4.2)

From (4.1) and (4.2) we see that $\varphi$ and therefore $\nu$ is
$\Gamma$-invariant.

###  4.3 Proof of Theorem D

By pulling back via a central extension, replacing $\Lambda$ with its preimage
and $\mu$ with an arbitrary lift (which is necessarily still a Furstenberg
measure), we assume as we may that $G$ is algebraically simply connected. We
may also assume that $G$ is connected due to Lemma 2.21. We consider an action
of $\Lambda$ on a finitely generated virtually nilpotent group $\Gamma$. By
Corollary 3.15 we assume as we may that $\Gamma$ is actually nilpotent. Recall
that for a finitely generated nilpotent group the set of torsion elements is a
characteristic finite subgroup. Applying again Corollary 3.15, we assume as we
may that $\Gamma$ is torsion free.

There exists a simply connected nilpotent Lie group $U$ that has $\Gamma$
embedded as a lattice [Rag72, Theorem 2.18]. The group of continuous
automorphisms $\operatorname{Aut}(U)$ is real algebraic subgroup of
$\operatorname{GL}\left(\operatorname{Lie}(U)\right)$.

If the action is finite, then the statement holds by Lemma 2.9. Otherwise, the
image of the action map $\rho:\Lambda\to\operatorname{Aut}(H)$ is unbounded in
$\operatorname{Aut}(U)$ because $\operatorname{Aut}(H)$ is discrete.
Margulis’s super-rigidity theorem [Mor01, Theorem 16.1.4] then guarantees a
finite index subgroup $\Lambda^{\prime}\leq\Lambda$ and a continuous
representation $\hat{\rho}:G\to\operatorname{Aut}(U)$ such that
$\hat{\rho}\\!\restriction_{\Lambda^{\prime}}=\rho$. By Theorem 4.1,
$\Lambda^{\prime}$ acts $\mu^{\prime}$-stiffly on
$\operatorname{Ch}\left(\Gamma\right)$ where $\mu^{\prime}$ is the hitting
measure coming from $\mu$. It follows by Lemma 2.21 that the action of
$\Lambda$ on $\operatorname{Ch}(\Gamma)$ is $\mu$-stiff.

###  4.4 The amenable radical of arithmetic groups

Let $\Lambda$ be an arithmetic subgroup of a $\mathbb{Q}$-algebraic group
$\mathbf{G}$. For a $\mathbb{Q}$-algebraic subgroup $\mathbf{H}$ of
$\mathbf{G}$ denote $\Lambda_{\mathbf{H}}:=\Lambda\cap\mathbf{H}(\mathbb{Q})$.
Then $\Lambda_{\mathbf{H}}$ is clearly an arithmetic subgroup of $\mathbf{H}$.
If $\bar{\mathbf{G}}$ is a quotient of $\mathbf{G}$ by a
$\mathbb{Q}$-algebraic normal subgroup, denote $\Lambda_{\bar{\mathbf{G}}}$
for the image of $\Lambda$ under the quotient map
$\mathbf{G}\to\bar{\mathbf{G}}$. Then $\Lambda_{\bar{\mathbf{G}}}$ is an
arithmetic subgroup of $\bar{\mathbf{G}}$ [PRR93, Theorem 4.1].

######  Lemma 4.8.

Let $\Lambda$ be an arithmetic subgroup of a $\mathbb{Q}$-algebraic semisimple
group $\mathbf{G}$. Then $\operatorname{Rad}(\Lambda)$ is finite, and it is
moreover trivial assuming that $\mathbf{G}$ has no finite normal subgroups,
and no $\mathbb{Q}$-simple $\mathbb{R}$-anisotropic factors.

###### Proof.

Assume first that $\mathbf{G}$ has no $\mathbb{Q}$-simple
$\mathbb{R}$-anisotropic factors, in which case $\Lambda$ is Zariski dense in
$\mathbf{G}(\mathbb{R})$ by Borel’s density theorem. Let $\mathbf{H}$ denote
the $\mathbb{R}$-Zariski closure of $\operatorname{Rad}(\Lambda)$ in
$\mathbf{G}$. Then $\mathbf{H}(\mathbb{R})$ is an amenable normal subgroup of
$\mathbf{G}(\mathbb{R})$ and thus must be compact [Zim13, §4.1]. But
$\operatorname{Rad}(\Lambda)$ is discrete and so it must be finite, making
$\mathbf{H}$ finite as well. Hence, if $\mathbf{G}$ is in addition assumed to
have no finite normal subgroups, then $\operatorname{Rad}(\Lambda)$ is
trivial.

Now allow $\mathbf{G}$ to have $\mathbb{Q}$-simple $\mathbb{R}$-anisotropic
factors. Let $\mathbf{G}_{c}$ denote the product of all
$\mathbb{R}$-anisotropic $\mathbb{Q}$-simple factors and denote
$\bar{\mathbf{G}}:=\mathbf{G}/\mathbf{G}_{c}$. Then the image $\bar{\Lambda}$
of $\Lambda$ under this quotient is an arithmetic subgroup of
$\bar{\mathbf{G}}$. By the previous paragraph,
$\operatorname{Rad}(\bar{\Lambda})$ is finite and therefore also
$\overline{\operatorname{Rad}(\Lambda)}$ as it is clearly contained in
$\operatorname{Rad}(\bar{\Lambda})$. But $\Lambda_{\mathbf{G}_{c}}$ is finite
as it is a discrete subgroup of the compact group
$\mathbf{G}_{c}(\mathbb{R})$, so it follows that $\operatorname{Rad}(\Lambda)$
is finite. ∎

We now consider a general arithmetic subgroup $\Lambda$ of some
$\mathbb{Q}$-algebraic group $\mathbf{G}$. Let $\mathbf{R}$ be the solvable
radical of $\mathbf{G}$, and consider the semisimple group
$\mathbf{S}=\mathbf{G}/\mathbf{R}$. Observe that the size of any finite normal
subgroup of $\mathbf{S}$ is bounded from above by
$|\mathbf{Z}|\cdot[\mathbf{S}:\mathbf{S}^{\circ}]$ where $\mathbf{S}^{\circ}$
is the identity connected component of $\mathbf{S}$ and $\mathbf{Z}$ is the
(finite) center of $\mathbf{S}^{\circ}$. It follows that $\mathbf{S}$ admits a
unique maximal finite normal subgroup, which coincides with the center when
$\mathbf{S}$ is connected. Let $\bar{\mathbf{S}}$ be the quotient of
$\mathbf{S}$ by all $\mathbb{R}$-anisotropic $\mathbb{Q}$-simple factors, as
well as by the remaining maximal finite normal subgroup. We consider the
arithmetic subgroup $\Lambda_{\mathbf{R}}$ of $\mathbf{R}$ and
$\Lambda_{\bar{\mathbf{S}}}$ of $\bar{\mathbf{S}}$.

######  Proposition 4.9.

Let $\Lambda$ be an arithmetic subgroup of $\mathbf{G}$. Then:

  1. 1.

$\Lambda_{\mathbf{R}}$ is a finite index subgroup of
$\operatorname{Rad}(\Lambda)$

  2. 2.

$\Lambda_{\bar{\mathbf{S}}}$ is equal to
$\Lambda/\operatorname{Rad}(\Lambda)$.

###### Proof.

It is clear that $\Lambda_{\mathbf{R}}$ is contained in
$\operatorname{Rad}(\Lambda)$.
$\Lambda_{\mathbf{S}}=\Lambda/\Lambda_{\mathbf{R}}$ is an arithmetic subgroup
of the semisimple group $\mathbf{S}$ so that
$\operatorname{Rad}\left(\Lambda_{\mathbf{S}}\right)$ is finite by Lemma 4.8.
Item 1 follows as $\operatorname{Rad}(\Lambda)/\Lambda_{\mathbf{R}}$ is
contained in the finite group
$\operatorname{Rad}\left(\Lambda/\Lambda_{\mathbf{R}}\right)=\operatorname{Rad}\left(\Lambda_{\mathbf{S}}\right)$.

For 2, consider the quotient map $\sigma:\mathbf{G}\to\bar{\mathbf{S}}$, and
let $K\leq\Lambda$ be the kernel of $\sigma\\!\restriction_{\Lambda}$.
$\sigma(\operatorname{Rad}(\Lambda))$ is a normal amenable subgroup of the
arithmetic subgroup $\Lambda_{\bar{\mathbf{S}}}=\sigma(\Lambda)$ of
$\bar{\mathbf{S}}$. By Lemma 4.8, $\sigma(\operatorname{Rad}(\Lambda))$ is
trivial meaning that $\operatorname{Rad}(\Lambda)\subseteq K$. But $K$ is
clearly normal and amenable and so we actually have
$\operatorname{Rad}(\Lambda)=K$. It follows that
$\Lambda/\operatorname{Rad}(\Lambda)=\Lambda_{\bar{\mathbf{S}}}$. ∎

As any solvable arithmetic group is polycyclic [Mal51] we get the following as
a consequence:

######  Corollary 4.10.

The amenable radical of an arithmetic group is virtually polycyclic.

###  4.5 Proof of Theorem C

The following lemma and its proof can be found in a more general form in
[CM84]:

######  Lemma 4.11 ([CM84, Lemma 2.2]).

Let $H$ be a discrete group, $N\leq H$ a normal subgroup, and assume that
$A:=H/N$ is abelian. Consider the continuous action of $\hat{A}$ on
$\operatorname{Ch}(H)$ given by pointwise multiplication:
$(\chi,\varphi)\mapsto\chi\cdot\varphi$. Then this action preserves the fibers
of the restriction map $\operatorname{Ch}(H)\to\operatorname{Ch}_{H}(N)$ and
acts on each of them transitively.

We first note that it is enough to show the statement of Theorem C for when
$\mathbf{G}$ is connected. The reason is that it is not hard to see that the
amenable radical of two commensurable subgroups is commensurable. Thus the
non-connected case is reduced to the connected case, using Lemma 2.21 and
Corollary 3.15.

Let $\mathbf{G}$ be a connected $\mathbb{Q}$-algebraic group with solvable
radical $\mathbf{R}$, unipotent radical $\mathbf{U}$ and semisimple part
$\mathbf{S}$. By general theory $\mathbf{R}/\mathbf{U}$ is abelian and is
centralized by $\mathbf{S}$. Let $\Lambda$ be an arithmetic subgroup of
$\mathbf{G}$ and let $\mu$ be a Furstenberg measure on $\Lambda$. We use the
notations introduced in §4.4. By the classical theorem of Borel and Harish-
Chandra $\Lambda_{\mathbf{S}}$ is a lattice in the real semisimple Lie group
$S=\mathbf{S}(\mathbb{R})$. We denote by $\bar{\mu}$ the measure on
$\Lambda_{\mathbf{S}}$ pushed from $\mu$. Clearly, this is a Furstenberg
measure. By Mal’cev theory, $\Lambda_{\mathbf{U}}$ is a lattice in the simply
connected nilpotent Lie group $U=\mathbf{U}(\mathbb{R})$. Thus, by Theorem
4.1, the action of $\Lambda_{\mathbf{S}}$ on
$\operatorname{Ch}(\Lambda_{\mathbf{U}})$ is stiff and by Proposition 3.14 so
is its action on
$\operatorname{Ch}_{\Lambda_{\mathbf{R}}}(\Lambda_{\mathbf{U}})$. Consider the
restricting map
$r:\operatorname{Ch}(\Lambda_{\mathbf{R}})\to\operatorname{Ch}_{\Lambda_{\mathbf{R}}}(\Lambda_{\mathbf{U}})$.
The group $A:=\Lambda_{\mathbf{R}}/\Lambda_{\mathbf{U}}$ is abelian and it is
centralized by $\Lambda_{\mathbf{S}}$. Hence by Lemma 4.11, the compact group
$\widehat{A}$ acts transitively on each fiber of $r$, commuting with the
action of $\Lambda_{\mathbf{S}}$. Using Proposition 2.15 with
$X=\operatorname{Ch}(\Lambda_{\mathbf{R}})$,
$Y=\operatorname{Ch}_{\Lambda_{\mathbf{R}}}(\Lambda_{\mathbf{U}})$ and
$K=\hat{A}$, we get that $r$ is relatively stiff with respect to the
$\Lambda_{\mathbf{S}}$-action. We deduce by Lemma 2.13 that the
$\Lambda_{\mathbf{S}}$-action on $\operatorname{Ch}(\Lambda_{\mathbf{R}})$ is
stiff and, using Lemma 2.10, we get that the $\Lambda$-action on
$\operatorname{Ch}(\Lambda_{\mathbf{R}})$ is stiff. Finally, using Proposition
4.9, we get by Corollary 3.15 that the $\Lambda$-action on
$\operatorname{Ch}(\operatorname{Rad}(\Lambda))$ is stiff.

##  5 Charmenability of arithmetic groups

###  5.1 A criterion for charmenability

######  Definition 5.1.

Let $\Lambda$ be a countable group and let $\Omega$ be a Lebesgue
$\Lambda$-space.

  1. 1.

$\Omega$ is called _metrically ergodic_ , if for any action of $\Lambda$ by
isometries on a separable metric space $Z$, any Lebesgue $\Lambda$-map
$\Omega\to Z$ is essentially constant.

  2. 2.

$\Omega$ is called _Zimmer amenable_ if for any Borel $\Lambda$-space $W$,
with an essentially surjective Borel $\Lambda$-map $p:W\to\Omega$ whose fibers
admit compact convex structures on which $\Lambda$ acts affinely, there exists
a Lebesgue $\Lambda$-map which is a section of $p$.

  3. 3.

$\Omega$ is called a _$\Lambda$ -boundary_ if it is both metrically ergodic
and Zimmer amenable.

The following is a strengthening of a theorem by Kaimanovich [Kai03].

######  Theorem 5.2 ([BF14, Theorem 2.7]).

Let $\Lambda$ be a countable group and let $\mu$ be an admissible probability
measure on $\Lambda$. Then the Furstenberg-Poisson boundary $\Pi(\Lambda,\mu)$
is a $\Lambda$-boundary.

We will make use of the following simple lemma:

######  Lemma 5.3.

Let $\Lambda$ be a countable group, let $N\lhd\Lambda$ be a normal amenable
subgroup and set $\Gamma=\Lambda/N$. If $B$ is a $\Gamma$-boundary then it is
also a $\Lambda$-boundary.

###### Proof.

We show that $B$ is metrically ergodic as a $\Lambda$-space assuming it is
such as a $\Gamma$-space (in fact regardless of whether $N$ is amenable), and
that $B$ is Zimmer amenable as a $\Lambda$-space assuming it is such as a
$\Gamma$-space.

Suppose $\Lambda$ acts by isometries on a separable metric space $Z$, and let
$f:B\to Z$ be a Lebesgue $\Lambda$-map. Since $N$ acts trivially on $B$, the
essential image of $f$ is supported on $N$-fixed points of $Z$. We may
therefore view $f$ as a Lebesgue $\Gamma$-map $B\to Z^{N}$. By metric
ergodicity of $B$ as a $\Gamma$-space, $f$ must be essentially constant. This
shows that $B$ is metrically ergodic as $\Lambda$-space

Consider now a Borel $\Lambda$-space $W$ with an essentially surjective Borel
$\Lambda$-map $p:W\to B$, whose fibers admit compact convex structures on
which $\Lambda$ acts affinely. Each fiber of $p$ is $N$-invariant, and
therefore admits $N$-fixed points. It follows that the map
$p\\!\restriction_{W^{N}}:W^{N}\to B$ is an essentially surjective Borel
$\Gamma$-map with compact convex fibers on which $\Gamma$ acts affinely. By
Zimmer amenability of $B$ as a Lebesgue $\Gamma$-space, there exists a
Lebesgue $\Gamma$-map which is a section of $p\\!\restriction_{W^{N}}$, and
this map is obviously also a Lebesgue $\Lambda$-map and a section of $p$. ∎

The proof of Theorem A is based on a criterion for charmenability established
in [BBHP22]. We shall first state it, and then explain the notions involved.

######  Criterion 5.4 ([BBHP22, Proposition 4.20]).

Let $\Lambda$ be a countable group and denote
$\Gamma:=\Lambda/\operatorname{Rad}(\Lambda)$. Assume that $\Gamma$ admits a
boundary $B$ satisfying the following two conditions:

  1. 1.

Every separable, ergodic, faithful $(\Gamma,L^{\infty}(B))$-von Neumann
algebra $(M,E)$ is either invariant or $\Gamma$-singular.

  2. 2.

Every Lebesgue $\Lambda$-map
$B\to\operatorname{PD}_{1}(\operatorname{Rad}(\Lambda))$ is essentially
constant.

Then $\Lambda$ is charmenable.

Note that when $\Lambda$ is arithmetic, the first condition of the criterion
deals only with the semisimple part. The fact that it holds in the higher rank
setting is essentially established in [BH21] and [BBHP22], see Proposition 5.7
bellow. As the main focus of this work is in cases where the amenable radical
is non-trivial, it is the second condition that is of our main interest. We
shall nevertheless briefly clarify the first condition, but emphasize that the
involved notions do not play an important conceptual role in this work. See
[BBHP22] for further details.

######  Definition 5.5.

Let $A$ be a C*-algebra, and let $\mathcal{S}(A)$ denote the space of states
on $A$. $\phi_{1},\phi_{2}\in\mathcal{S}(A)$ are called _singular_ if
$\|\phi_{1}-\phi_{2}\|=2$.

Given a Lebesgue space $\Omega$, two Lebesgue maps
$\theta_{1},\theta_{2}:\Omega\to\mathcal{S}(A)$ are called _singular_ if
$\theta_{1}(\omega)$ and $\theta_{2}(\omega)$ are singular for a.e
$\omega\in\Omega$.

Two unital completely-positive maps $E_{1},E_{2}:A\to L^{\infty}(\Omega)$ are
called _singular_ if the corresponding dual maps
$\theta_{1},\theta_{2}:\Omega\to\mathcal{S}(A)$ (given by
$\theta_{i}(\omega)(a)=E_{i}(a)(\omega)$) are singular.

Given a separable von Neumann algebra $M$, two normal unital completely-
positive maps $E_{1},E_{2}:M\to L^{\infty}(\Omega)$ are called _singular_ if,
$M$ admits a separable unital C*-subalgebra $A$ such that the restrictions
$E_{1}\\!\restriction_{A},E_{2}\\!\restriction_{A}$ are singular.

######  Definition 5.6.

Let $\Lambda$ be a countable group.

A _$\Lambda$ -von Neumann algebra_ is a von Neumann algebra $N$ endowed with a
ultraweak continuous action of $\Lambda$ by automorphisms. $N$ is said to be
_ergodic_ if the subalgebra of fixed point $N^{\Lambda}$ is trivial (i.e
$=\mathbb{C}$).

Given a $\Lambda$-von Neumann algebra $N$, a _$(\Lambda,N)$ -von Neumann
algebra_ is a pair $\left(M,E\right)$ where $M$ is a $\Lambda$-von Neumann
algebra, and $E$ is a unital completely-positive $\Lambda$-equivariant map
$E:M\to N$. $(M,E)$ is said _faithful_ or _extremal_ if $E$ satisfies the
corresponding properties. If $E(M)\subset N^{\Lambda}$, we say that $(M,E)$ is
_ $\Lambda$-invariant_.

Assume $N=L^{\infty}(\Omega)$ where $\Omega$ is a Lebesgue $\Lambda$-space.
Then $E$ is said to be _$\Lambda$ -singular_ if the normal unital completely-
positive maps $E_{g}:x\in M\mapsto E(gx)\in L^{\infty}(\Omega)$ are pairwise
singular.

The only important example for our concern is when $N$ is the commutative von
Neumann algebra of $L^{\infty}$-functions on the Furstenberg-Poisson boundary
of $\Lambda$ w.r.t some measure $\mu$. In such case $(\Lambda,N)$-von Neumann
algebras may be seen as a non-commutative version of the boundary maps
appearing in Theorem 2.17.

###  5.2 Proof of Theorem A

The first condition of Criterion 5.4 was essentially shown to hold for
arithmetic subgroups of semisimple groups of higher rank:

######  Proposition 5.7.

Let $\Gamma$ be an arithmetic subgroup of a connected $\mathbb{Q}$-algebraic
group $\mathbf{G}$, and let $B$ be its Poisson boundary w.r.t a Furstenberg
measure $\mu$. Assume that $\mathbf{G}$ is $\mathbb{Q}$-simple, center-free,
without $\mathbb{R}$-anisotropic $\mathbb{Q}$-simple factors, and with
$\mathbb{R}$-rank at least $2$. Then the first condition of Criterion 5.4
holds true.

###### Proof.

Let $G=\mathbf{G}(\mathbb{R})$. By Lemma 4.8, $\operatorname{Rad}(\Gamma)$ is
trivial. The intersection of $\Gamma$ with any compact simple factor of $G$ is
central, hence trivial. We may therefore assume that $G$ has no compact simple
factors. Let $(M,E)$ be a separable, ergodic, faithful
$(\Gamma,L^{\infty}(B))$-von Neumann algebra which is not $\Gamma$-invariant.
We explain how to deduce singularity of $E$ from the work of [BH21] or of
[BBHP22], depending on whether $G$ is simple or a product of a few simple
factors.

Assume first $G$ is simple. Let $P$ be a minimal parabolic subgroup of $G$ so
that $B=G/P$. Then by [BH21, Theorem B] there is a proper closed subgroup
$P\leq Q\lneq G$ and a unital normal $\Gamma$-equivariant $*$-embedding
$\iota:L^{\infty}(G/Q)\to M$ such that $E\circ\iota$ is equal to the natural
embedding $p_{Q}^{*}:L^{\infty}(G/Q)\to L^{\infty}(G/P)$. Note that the
theorem in this reference is formalized in a somewhat dual way, though it is
equivalent (see [Hou21, Theorem E]). Let $M_{0}=\iota(L^{\infty}(G/Q))$ and
consider $(M_{0},E\\!\restriction_{M_{0}})$ as a $(G,L^{\infty}(G/P))$-von
Neumann algebra. It is not $G$-invariant as $Q$ is proper, and it is extremal
because $M_{0}$ is ergodic (see [BBHP22, Lemma 4.16]). Due to [BBHP22, Example
2.14, Lemma 6.2], the singularity criterion of [BBHP22, Proposition 4.15]
applies so that $(M_{0},E\\!\restriction_{M_{0}})$ is $G$-singular. It is
therefore in particular $\Gamma$-singular. Since $M$ contains $M_{0}$ as a
subalgebra, it follows that $(M,E)$ is singular as well (see [BBHP22, Remark
4.12]).

Now assume $G$ has at least two (non-compact) simple factors. Then, in the
terminology of [BBHP22], $\Gamma$ is an arithmetic group of product type and
is therefore charmenable by [BBHP22, Proposition 6.1]. More specifically, it
is transparent that the proof of this proposition all boils down to showing
that $E$ and $E_{\gamma}$ are singular for any non-central element
$\gamma\in\Gamma$. But in our case $\Gamma$ is center-free, and so $E$ is
$\Gamma$-singular. ∎

The dynamical study of §4, which eventually lead to the proof of Theorem C,
was aimed towards the establishment of the second condition of Criterion 5.4
for arbitrary arithmetic groups:

######  Proposition 5.8.

Let $\Lambda$ be an arithmetic group and let $(B,\nu_{B})$ be its Poisson
boundary w.r.t a Furstenberg measure $\mu$. Then the second condition of
Criterion 5.4 holds true.

###### Proof.

Let $\theta:B\to\operatorname{PD}_{1}(\operatorname{Rad}(\Lambda))$ be a
Lebesgue $\Lambda$-map. Since $\operatorname{Rad}(\Lambda)$ acts trivially on
$B$, the essential image of $\theta$ consists of traces, namely, we may view
$\theta$ as a Lebesgue $\Lambda$-map
$\theta:B\to\operatorname{Tr}(\operatorname{Rad}(\Lambda))$. By pushing
forward $\nu_{B}$ through $\theta$ and taking its barycenter, we get a
$\mu$-stationary trace
$\varphi\in\operatorname{Tr}(\operatorname{Rad}(\Lambda))^{\mu}$. The
$\mu$-stationary probability measure on
$\operatorname{Ch}(\operatorname{Rad}(\Lambda))$ corresponding to $\varphi$
(Theorem 3.1) is $\Lambda$-invariant by Theorem C. It follows that $\varphi$
is $\Lambda$-invariant, which means that $\theta$ is essentially constant by
Lemma 2.18. ∎

###### Proof of Theorem A.

Let $\mathbf{G}$ be a $\mathbb{Q}$-algebraic group with solvable radical
$\mathbf{R}$ and $\mathbf{S}=\mathbf{G}/\mathbf{R}$, and let
$\Lambda_{\mathbf{S}}$ be the image of the arithmetic subgroup $\Lambda$ under
the quotient map $\mathbf{G}\to\mathbf{S}$. We first observe that if
$\mathbf{S}$ has more than one $\mathbb{R}$-isotropic $\mathbb{Q}$-simple
factor then $\Lambda_{\mathbf{S}}$ has a normal subgroup which is neither
amenable nor co-amenable. Also, if the real rank of $\mathbf{S}$ is $1$ then
$\Lambda_{\mathbf{S}}$ has such a normal subgroup, as it is relatively
hyperbolic. In both cases we see that $\Lambda$ admits a normal subgroup which
is neither amenable nor co-amenable which implies that $\Lambda$ is not
charmenable [BBHP22, Propositions 3.3]. This shows that the conditions imposed
on $\mathbf{S}$ in the statement of the theorem are necessary for
charmenability of $\Lambda$. We now turn to show that they are sufficient as
well. Note that as $\Lambda$ is an arbitrary arithmetic subgroup of
$\mathbf{G}$, this will in particular show that either all arithmetic
subgroups of $\mathbf{G}$ are charmenable, or else none of them are.

If the real rank of $\mathbf{S}$ is $0$ then $\mathbf{S}(\mathbb{R})$ is
compact in which case $\mathbf{G}(\mathbb{R})$ is amenable, hence $\Lambda$ is
amenable and a fortiori charmenable. We thus assume from now on that
$\mathbf{S}$ has at most one $\mathbb{R}$-isotropic $\mathbb{Q}$-simple factor
and that the real rank of $\mathbf{S}$ is at least $2$, and we argue to show
that $\Lambda$ is charmenable. We will do so by showing that it satisfies the
first condition of Criterion 5.4 (the second condition holds in general by
Proposition 5.8)

Let $\bar{\mathbf{S}}$ denote the quotient of $\mathbf{S}$ by all
$\mathbb{R}$-anisotropic $\mathbb{Q}$-simple factors, as well as by the
remaining center. Let $\Lambda_{\bar{\mathbf{S}}}$ denote the image of
$\Lambda$ under the quotient map $\mathbf{G}\to\bar{\mathbf{S}}$. Fix a
Furstenberg measure $\mu$ on $\Lambda$ and let $\bar{\mu}$ be the pushforward
measure onto $\Lambda_{\bar{\mathbf{S}}}$, which is a Furstenberg measure as
well. The Furstenberg-Poisson boundary of $\Lambda$ w.r.t $\mu$ is the same as
the Furstenberg-Poisson boundary $\Lambda_{\bar{\mathbf{S}}}$ w.r.t
$\bar{\mu}$ and we denote it by $B$. By Theorem 5.2, $B$ is a
$\Lambda_{\bar{\mathbf{S}}}$-boundary and therefore also a $\Lambda$-boundary
by Lemma 5.3. By Proposition 4.9, we have that
$\Lambda/\operatorname{Rad}(\Lambda)=\Lambda_{\bar{\mathbf{S}}}$. In
particular, this is an arithmetic subgroup of $\bar{\mathbf{S}}$. This first
condition of Criterion 5.4 therefore follow from Proposition 5.7. ∎

##  6 Properties of charmenable groups

This section is dedicated for proving Propositions 1.3 and 1.4 from the
introduction. Here too, $\Lambda$ denotes a countable group.

###  6.1 Amenable traces

Let $\pi:\Lambda\to\mathcal{U}(\mathcal{H})$ be a representation. A sequence
of vectors $v_{n}\in\mathcal{H}$ of norm $1$ is said to be _almost
$\Lambda$-invariant_ if:

$\forall g\in\Lambda:\quad\|\pi(g)v_{n}-v_{n}\|\to 0$

If such a sequence exists, we say that $\pi$ has _almost invariant vectors_.
The representation $\pi$ is called _amenable_ if $\pi\otimes\bar{\pi}$ has
almost invariant vectors, where $\bar{\pi}$ is the dual representation. See
[Bek90] for more on this notion, including properties and equivalent
definitions. A trace $\varphi$ on $\Lambda$ is called _amenable_ if its GNS
representation is amenable. This should not be confused with the stronger
notion of a von Neumann amenable trace given in Definition 3.10. The second
condition in the definition of charmenability implies the dichotomy stated on
traces:

######  Proposition 6.1 ([BBHP22, Proposition 3.2]).

Assume that every character of $\Lambda$ is either von Neumann amenable, or
supported on $\operatorname{Rad}{(\Lambda)}$. Then any trace of $\Lambda$ is
either amenable or supported on $\operatorname{Rad}({\Lambda})$.

Note that a representation $\pi$ of a product of groups
$\Gamma_{1}\times\Gamma_{2}$ may have no almost invariant vectors even though
both restrictions $\pi\\!\restriction_{\Gamma_{1}}$,
$\pi\\!\restriction_{\Gamma_{2}}$ have. Indeed, if $\pi_{i}$ is a
representation of $\Gamma_{i}$ without almost invariant vectors then we may
extend each $\pi_{i}$ to $\Gamma_{1}\times\Gamma_{2}$ trivially, and consider
$\pi=\pi_{1}\oplus\pi_{2}$. The following proposition shows that this is the
only obstruction.

######  Proposition 6.2.

Suppose that $\Gamma_{1},\Gamma_{2}$ are two subgroups of $\Lambda$ which
commute with each other and satisfy $\Lambda=\Gamma_{1}\Gamma_{2}$. Let
$\pi:\Lambda\to\mathcal{U}(\mathcal{H})$ be a representation of $\Lambda$.
Assume that the restrictions $\pi\\!\restriction_{\Gamma_{1}}$ and
$\pi\\!\restriction_{\Gamma_{2}}$ admit almost invariant vectors, whereas
$\pi$ does not. Then $\pi$ decomposes as a direct sum
$\pi=\pi_{1}\oplus\pi_{2}$ such that $\pi_{1}\\!\restriction_{\Gamma_{1}}$ and
$\pi_{2}\\!\restriction_{\Gamma_{2}}$ admit almost invariant vectors but
$\pi_{1}\\!\restriction_{\Gamma_{2}}$ and
$\pi_{2}\\!\restriction_{\Gamma_{1}}$ do not.

###### Proof.

We first recall a general fact about almost invariant vectors. Consider a
countable group $\Gamma$ and fix an admissible probability measure $\mu$ on
$\Gamma$ satisfying $\mu(\gamma^{-1})=\mu(\gamma)$ for all $\gamma\in\Gamma$.
Given a unitary representation $\rho:\Gamma\to\mathcal{U}(\mathcal{H})$, there
is the associated Markov operator:

$\mathcal{B}(\mathcal{H})\ni
A_{\mu}:\,\,\,\,\,\,\,v\mapsto\sum_{\gamma\in\Gamma}\mu(\gamma)\rho(\gamma)v$

Then $A_{\mu}$ is a positive contractive operator, and its norm is $1$ if and
only if $\rho$ has almost invariant vectors [Pet13, Proposition 7.1.4]. We
apply this reasoning to $\Gamma_{1}$ and $\Gamma_{2}$, that is we fix
generating probability measures on each of those subgroups, and consider the
corresponding Markov operators denoted by $A_{1}$ and $A_{2}$. Then $1$ is in
the spectrum of both $A_{1}$ and $A_{2}$, and yet, there is some $\epsilon>0$
such that:

$\max_{i=1,2}\|A_{i}v-v\|>\epsilon,\quad\forall v\in\mathcal{H}\,\text{ with
}\,||v||=1$ (6.1)

For $i=1,2$, let $p_{i}\in\mathcal{B}(\mathcal{H})$ denote the spectral
projection of $A_{i}$ associated with the interval $[1-\epsilon,\epsilon]$,
and denote by $\mathcal{K}_{i}:=p_{i}\mathcal{H}\subseteq\mathcal{H}$ the
corresponding subspace. Then $\mathcal{K}_{i}$ is an $A_{i}$-invariant
subspace which contains almost $\Gamma_{i}$-invariant vectors, whereas its
orthogonal complement $\mathcal{K}_{i}^{\bot}$ does not. Moreover, it follows
from Eq. (6.1) that $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$ are orthogonal
subspaces i.e $\mathcal{K}_{2}\subseteq\mathcal{K}_{1}^{\bot}$. Finally note
that $\mathcal{K}_{1}$ is $\Gamma_{2}$-invariant (and vice versa). Indeed,
this can be seen as a consequence of the fact that von Neumann algebras are
closed under taking spectral projections, so that $p_{1}$ is in the double
commutant of $A_{1}$ and thus commutes with $\pi(\Gamma_{2})$.

Thus for each $\gamma_{1}\in\Gamma_{1}$,
$\pi(\gamma_{1})\mathcal{K}_{1}^{\bot}$ is a $\Gamma_{2}$-invariant subspace
that contains $\mathcal{K}_{2}$. The intersection
$\mathcal{H}_{2}:=\bigcap_{\gamma_{1}\in\Gamma_{1}}\pi(\gamma_{1})\mathcal{K}_{1}^{\bot}$
is therefore a $\Gamma_{1}\times\Gamma_{2}$-invariant subspace which contains
$\mathcal{K}_{2}$, and as a result admits almost $\Gamma_{2}$-invariant
vectors. $\mathcal{H}_{2}$ however does not admit almost
$\Gamma_{1}$-invariant vectors because it is contained in
$\mathcal{K}_{1}^{\bot}$. The orthogonal complement
$\mathcal{H}_{1}:=\mathcal{H}_{2}^{\bot}$ on the other hand has almost
$\Gamma_{1}$-invariant vectors because it contains $\mathcal{K}_{1}$, but does
not have almost $\Gamma_{2}$-invariant vectors because it is contained in
$\mathcal{K}_{2}^{\bot}$. The subrepresentations $\pi_{1},\pi_{2}$
corresponding to the invariant subspaces $\mathcal{K}_{1}$,$\mathcal{K}_{2}$
therefore give the desired decomposition. ∎

Recall that to any positive-definite function $\varphi$ corresponds a GNS
triplet $(\mathcal{H},\pi,\xi)$. If $\varphi$ is moreover a trace then $\pi$
extends to a representation
$\pi\times\rho:\Lambda\times\Lambda\to\mathcal{U}(\mathcal{H})$ such that
$\pi(g)\xi=\rho(g^{-1})\xi$ for all $g\in G$, see Example 3.17. In particular
$\varphi(g)=\langle\pi(g)\xi,\xi\rangle=\langle\xi,\rho(g)\xi\rangle$. We call
$\rho$ the _right GNS representation of $\varphi$_. To avoid confusion, we
shall call $\pi$ the _left GNS representation_.

######  Proposition 6.3.

Let $\varphi$ be a trace of $\Lambda$, and let $\pi$ and $\rho$ be the left
and right GNS representations of $\varphi$. If $\pi$ admits almost invariant
vectors then so does
$\pi\times\rho:\Lambda\times\Lambda\to\mathcal{U}(\mathcal{H})$.

###### Proof.

Let $(\mathcal{H},\pi,\xi)$ be a GNS triplet associated with $\varphi$.
Observe that:

$\varphi(g)=\left\langle\xi,\rho(g)\xi\right\rangle=\left\langle\rho(g)^{*}\xi,\xi\right\rangle,\quad
g\in\Lambda$ (6.2)

Thus $\pi$ is equivalent to the adjoint representation $\bar{\rho}$, by the
uniqueness of the GNS. Since $\pi$ admits almost invariant vectors, then so
does $\rho$. Assuming by contradiction that $\pi\times\rho$ does not admit
almost invariant vectors implies, by Proposition 6.2, that there is a
subrepresentation $\pi_{1}\times\rho_{1}$ of $\pi\times\rho$ such that
$\pi_{1}$ has almost invariant vectors but $\rho_{1}$ does not. Let
$\mathcal{H}_{1}$ denote the subspace corresponding to this subrepresentation.
Since $\xi$ is cyclic with respect to $\pi$ as well as to $\rho$, its
projection onto $\mathcal{H}_{1}$ is non-zero and we denote its normalization
by $\xi_{1}\in\mathcal{H}_{1}$. It is not hard to see that $\xi_{1}$ must be
cyclic with respect to the representations $\pi_{1}$ and $\rho_{1}$.

Let $\varphi_{1}$ denote the positive-definite function corresponding to the
GNS triplet $(\mathcal{H}_{1},\pi_{1},\xi_{1})$. Then a similar computation to
Eq. (6.2) shows that $(\mathcal{H}_{1},\bar{\rho}_{1},\xi_{1})$ is a GNS
triplet associated with $\varphi_{1}$. By uniqueness of the GNS we conclude
that $\pi_{1}$ and $\bar{\rho}_{1}$ are equivalent, and in particular
$\rho_{1}$ admits almost invariant vectors. This is a contradiction. ∎

######  Proposition 6.4.

Let $\varphi$ be a trace of $\Lambda$, and let $\pi$ and $\rho$ be the left
and right GNS representations corresponding to $\varphi$. If $\varphi$ is
amenable then $\pi\times\rho:\Lambda\times\Lambda\to\mathcal{U}(\mathcal{H})$
is amenable.

###### Proof.

The function $|\varphi|^{2}=\varphi\cdot\bar{\varphi}$ is a trace on
$\Lambda$. Indeed,
$\left(\mathcal{H}\otimes\mathcal{H},\pi\otimes\bar{\pi},\xi\otimes\xi\right)$
is the corresponding GNS triplet, and $\rho\otimes\bar{\rho}$ is the right
GNS. As $\varphi$ is amenable, $\pi\otimes\bar{\pi}$ has almost invariant
vectors, and so by Proposition 6.3 so does
$\left(\pi\otimes\bar{\pi}\right)\times\left(\rho\otimes\bar{\rho}\right)$.
But the latter is canonically equivalent to
$\left(\pi\times\rho\right)\otimes\overline{\left(\pi\times\rho\right)}$ so
that $\pi\times\rho$ is amenable. ∎

###  6.2 Invariant random subgroups and p.m.p actions

For a group $\Lambda$ denote by $\operatorname{Sub}({\Lambda})$ the space of
all subgroups of $\Lambda$ endowed with the Chabauty topology. This topology
is the one induced by the natural embedding
$\operatorname{Sub}({\Lambda})\subseteq\\{0,1\\}^{\Lambda}$ with the product
topology, and in particular it is compact. We say that
$H\in\operatorname{Sub}{(\Lambda)}$ is _co-amenable_ if $\Lambda/H$ admits a
$\Lambda$-invariant mean (see [HT16]).

######  Lemma 6.5.

The set of co-amenable subgroups of a countable group $\Lambda$ is Borel.

###### Proof.

Let $F_{k}$ be an enumeration of all finite subsets of $\Lambda$. For a
subgroup $H\leq\Lambda$ and a set $S\subseteq\Lambda$ let $S/H$ denote the
image of $S$ under the quotient map $\Lambda\to\Lambda/H$. Clearly,
$\left\\{F_{k}/H\right\\}_{k\in\mathbb{N}}$ exhausts all possible finite
subsets of $\Lambda/H$. For $g\in\Lambda$ and $k\in\mathbb{N}$ define the map
$f_{k,g}:\operatorname{Sub}({\Lambda})\to\mathbb{Q}$ defined by:

$f_{k,g}\left(H\right)=\frac{\left|\left(F_{k}\bigtriangleup
g.F_{k}\right)/H\right|}{\left|F_{k}/H\right|}$

Suppose a sequence of subgroups $H_{n}$ converges to $H$. Since $\Lambda$ is
discrete, this just means that a given $x\in\Lambda$ is in $H$ if and only if
it is eventually in $H_{n}$. Therefore, $\left|S/H_{n}\right|$ eventually
coincides with $|S/H|$ for any given finite set $S\subseteq\Lambda$. This
shows that $f_{k,g}$ is locally constant and in particular continuous. It
follows that the set

$\bigcap_{i,j\in\mathbb{N}}\bigcup_{k\in\mathbb{N}}\bigcap_{g\in
F_{i}}f_{k,g}^{-1}\left([0,2^{-j})\right)$

is Borel. It is left to note that this set is precisely the set of all co-
amenable subgroups, see the Følner condition for co-amenability [GM07, Theorem
2.9]. ∎

We consider the continuous action of $\Lambda$ on
$\operatorname{Sub}(\Lambda)$ by conjugation. An _invariant random subgroup_
(IRS in short) of $\Lambda$ is a $\Lambda$-invariant Borel probability measure
on $\operatorname{Sub}(\Lambda)$. A _uniformly recurrent subgroup_ (URS in
short) is a minimal (non-empty) $\Lambda$-invariant closed subset of
$\operatorname{Sub}({\Lambda})$. We refer to [AGV14, GW15] for more on these
notions.

By a _p.m.p action_ we mean a probability measure space $(\Omega,\nu)$ endowed
with a measure preserving action of $\Lambda$. We say that such an action has
a _spectral gap_ if the associated Koopman representation
$\Lambda\to\mathcal{U}(L^{2}_{0}(\Omega,\nu))$ (where $L^{2}_{0}$ is the
orthogonal complement of the constants) has no almost invariant vectors. We
note that having a spectral gap implies ergodicity. The pushforward of $\nu$
under the stabilizer map $\omega\mapsto\mathrm{stab}_{\Lambda}(\omega)$ is an
IRS on $\Lambda$. In fact, any IRS arises in this manner, and any ergodic IRS
arises from an ergodic p.m.p action [AGV14].

######  Theorem 6.6 (Compare with [PT16, Theorem 3.2] and with [DM14, Theorem
2.12]).

Let $\Lambda$ be a countable group, and suppose that any trace of $\Lambda$ is
either amenable or supported on the amenable radical. Then for any p.m.p
action $(\Omega,\nu)$ of $\Lambda$ with spectral gap, the stabilizer of a
random point $\omega\in\Omega$ is either contained in
$\operatorname{Rad}(\Lambda)$ almost surely, or it is co-amenable almost
surely. Thus, denoting by $\mu$ the corresponding IRS, either the set of all
subgroups of $\operatorname{Rad}(\Lambda)$ is of full $\mu$-measure, or the
set of all co-amenable subgroups is of full $\mu$-measure.

We shall first need some preparation.

######  Lemma 6.7.

Consider a measure space $(\Omega,\nu)$ and a measure preserving action of a
group $\Lambda$ on it, and fix $1\leq p<\infty$. Let $\mathcal{F}_{p}$ denote
the set of functions $\Omega\to\mathbb{R}_{\geq 0}$ in the unit sphere of
$L^{p}(\Omega,\nu)$. Let $\mathcal{P}$ denote the set of probability measure
on $\Omega$ which are absolutely continuous w.r.t $\nu$, endowed with the
total variation metric. Then the mapping:

$\phi_{p}:\mathcal{F}_{p}\to\mathcal{P}:f\mapsto f^{p}\nu$

is a $\Lambda$-equivariant bijection, and both $\phi_{p}$ and $\phi^{-1}_{p}$
are uniformly continuous. In particular, for a sequence
$f_{n}\in\mathcal{F}_{p}$ the following are equivalent:

  1. 1.

$f_{n}$ are almost invariant in the $L^{p}$-norm:

$\forall g\in\Lambda:\|g.f_{n}-f_{n}\|_{p}\to 0$

  2. 2.

$\nu_{n}=\phi_{p}(f_{n})$ are almost invariant in the total variation norm:

$\forall g\in\Lambda:\|g.\nu_{n}-\nu_{n}\|_{\mathrm{tot}}\to 0$

###### Proof.

Consider first the map $\mathcal{F}_{p}\to\mathcal{F}_{1}:f\mapsto f^{p}$.
This map is referred to in the literature as the Mazur map, and it is well
known that both this map, as well as its inverse, are uniformly continuous
[BL98, Chapter 9.1]. It is also clearly a $\Lambda$-equivariant bijection. We
may therefore assume that $p=1$ and consider the map $\phi_{1}$.

$\phi_{1}$ is of course bijective and it is evidently $\Lambda$-equivariant.
It is a standard fact that the total variation distance between two absolutely
continuous probability measures is precisely half of the $L^{1}$-distance of
the corresponding probability density functions, so in particular, $\phi_{1}$
and $\phi_{1}^{-1}$ are uniformly continuous.

The equivalence between almost invariance of the $L^{p}$-functions and the
corresponding measures follows from equivariance and uniform continuity.

∎

Let $\Lambda\curvearrowright\left(\Omega,\nu\right)$ be a p.m.p action. We
define $\varphi:\Lambda\to\mathbb{C}$ by
$\varphi(g)=\nu\left(\operatorname{Fix}({g})\right)$ where
$\operatorname{Fix}(g)=\left\\{\omega\in\Omega\mid g.\omega=\omega\right\\}$,
and note that this is a trace on $\Lambda$. We recall a construction of a
representation associated with this p.m.p action that utilizes the GNS
representation of $\varphi$. Consider the orbit equivalence relation:

$\mathcal{R}:=\left\\{\left(g.\omega,\omega\right)\mid\omega\in\Omega,g\in\Lambda\right\\}\subseteq\Omega\times\Omega$

endowed with the subspace $\sigma$-algebra of the product. Denote by $p_{1}$
the projection of $\mathcal{R}$ onto the first coordinate, and define the
measure $\tilde{\nu}$ on $\mathcal{R}$ by setting for any measurable
$E\subseteq\mathcal{R}$:

$\tilde{\nu}\left(E\right)=\int_{\Omega}\left|p_{1}^{-1}(\omega)\cap
E\right|d\nu$

In other words, $\tilde{\nu}$ is obtained by integrating the counting measures
on the fibers of $p_{1}$ along $\nu$.

$\Lambda$ acts on $\mathcal{R}$ via the left coordinate, as well as via the
right coordinate, and consider the corresponding Koopman representations
$\pi,\rho:\Lambda\to\mathcal{U}\left(L^{2}(\mathcal{R},\tilde{\nu})\right)$.
$\pi$ and $\rho$ obviously commute with each other and satisfy
$\pi(g)\rho(g)\boldsymbol{1}_{\Delta}=\boldsymbol{1}_{\Delta}$ where
$\boldsymbol{1}_{\Delta}$ is the characteristic function of the diagonal of
$\Omega\times\Omega$. Moreover
$\varphi(g)=\left\langle\pi(g)\boldsymbol{1}_{\Delta},\boldsymbol{1}_{\Delta}\right\rangle$
where $\boldsymbol{1}_{\Delta}$ is the characteristic function of the diagonal
of $\Omega\times\Omega$. This means the left and right GNS representations of
$\varphi$ are the cyclic subrepresentation of $\pi$ generated by
$\boldsymbol{1}_{\Delta}$. See [BdlH19, Theorem 15.F.4] for further details.

######  Lemma 6.8.

Let $(\Omega,\nu)$ be a p.m.p action and let
$\varphi:g\mapsto\nu(\operatorname{Fix}(g))$ be the corresponding trace of
$\Lambda$. If $(\Omega,\nu)$ has a spectral gap and $\varphi$ is amenable then
the stabilizer of $\nu$-a.e point in $\Omega$ is co-amenable.

###### Proof.

Consider the representation
$\pi\times\rho:\Lambda\times\Lambda\to\mathcal{U}(L^{2}(\mathcal{R},\tilde{\nu}))$
where $(\mathcal{R},\tilde{\nu})$ is the orbit equivalence relation space
associated with $(\Omega,\nu)$, as constructed above. By Proposition 6.4, the
combined left and right GNS representation $\pi_{\varphi}\times\rho_{\varphi}$
of $\varphi$ is amenable. Hence $\pi\times\rho$ is amenable as it contains
$\pi_{\varphi}\times\rho_{\varphi}$. This means that there are functions
$f_{n}\in L^{2}(\mathcal{R})^{\otimes 2}\subseteq L^{2}(\mathcal{R}^{2})$ of
norm $1$ which are almost invariant for the $\Lambda\times\Lambda$-action. In
particular $|f_{n}|$ are almost invariant and so by Lemma 6.7, the probability
measures $|f_{n}|^{2}\tilde{\nu}^{2}$ on $\mathcal{R}^{2}$ are almost
invariant (in the total variation norm). Thus so are the pushforward
probability measure $\eta_{n}$ under the projection
$\mathcal{R}^{2}\to\mathcal{R}$ (onto say, the first coordinate).

The projection $p_{1}:\mathcal{R}\to\Omega$ onto the first coordinate is
$\Lambda\times\\{e\\}$-equivariant. Thus the pushforward probability measures
$\nu_{n}:={p_{1}}_{*}\eta_{n}$ on $\Omega$ are absolutely continuous w.r.t
$\nu$ and are almost invariant. By slightly changing $\nu_{n}$ (e.g taking a
convex combination with $\nu$) we may assume that they are all in the same
measure class of $\nu$. Using Lemma 6.7 once again, we deduce that the
$L^{2}$-functions $\sqrt{{d\nu_{n}}/{d\nu}}$ are almost invariant. By