# Infinite Volume and Infinite Injectivity Radius

Mikolaj Fraczyk and Tsachik Gelander

###### Abstract.

We prove the following conjecture of Margulis. Let $G$ be a higher rank simple
Lie group and let $\Lambda\leq G$ be a discrete subgroup of infinite covolume.
Then, the locally symmetric space $\Lambda\backslash G/K$ admits injected
balls of any radius. This can be considered as a geometric interpretation of
the celebrated Margulis normal subgroup theorem. However, it applies to
general discrete subgroups not necessarily associated to lattices. Yet, the
result is new even for subgroups of infinite index of lattices. We establish
similar results for higher rank semisimple groups with Kazhdan’s property (T).
We prove a stiffness result for discrete stationary random subgroups in higher
rank semisimple groups and a stationary variant of the Stück–Zimmer theorem
for higher rank semisimple groups with property (T). We also show that a
stationary limit of a measure supported on discrete subgroups is almost surely
discrete.

##  1\. Introduction

###  1.1. The statement for simple groups

The main motivation for this paper was to prove the following result which
confirms a conjecture of G.A. Margulis:

######  Theorem 1.1.

Let $G$ be a connected centre-free simple Lie group of real rank at least $2$
and let $X=G/K$ be the associated symmetric space. Let $\Lambda\leq G$ be a
discrete group of infinite covolume. Then for every $r>0$ there is a point
$p\in\Lambda\backslash X$ where the injectivity radius is at least $r$.

Theorem 1.1 can be regarded as a geometric interpretation of Margulis’ normal
subgroup theorem for lattices in $G$. In particular the analogous result is
false if $G$ has rank one.

######  Example 1.2.

Let $G$ be a simple group of rank one, let $\Gamma\leq G$ be a uniform lattice
and let $\Delta\lhd\Gamma$ be a nontrivial normal subgroup of infinite index.
The infinite volume locally symmetric space $M=\Delta\backslash G/K$ has
bounded injectivity radius. Indeed, let $\alpha\in\Delta\setminus\\{1\\}$, let
$\Omega$ be a compact fundamental domain for $\Gamma$ in $G/K$ and let
$D=\max\\{d(x,\alpha\cdot x):x\in\Omega\\}$. Then $\text{InjRad}_{M}(p)\leq
D/2$ at any point $p\in M$. To see this consider a lift $\tilde{p}\in G/K$ and
let $\gamma\in\Gamma$ be an element such that
$\gamma^{-1}\cdot\tilde{p}\in\Omega$. Then the element
$\gamma\alpha\gamma^{-1}$ is in $\Delta$ and has displacement at most $D$ at
$\tilde{p}$.

We prove Theorem 1.1 by showing that for a certain bi-$K$-invariant
probability measure $\nu=\nu_{G}$ on $G$ (see §2) the random walk on
$\Lambda\backslash X$ eventually spends most of the time in the $r$-thick part
for every $r$.

######  Theorem 1.3.

Let $M=\Lambda\backslash X$ be an $X$-orbifold of infinite volume. Let
$x_{0}\in X$ be an arbitrary point, set
$\mu_{n}=\frac{1}{n}\sum_{i=0}^{n-1}\nu^{(i)}*\delta_{x_{0}}$ and let
$\overline{\mu_{n}}$ be the pushforward of $\mu_{n}$ to $M$ via the covering
map. For every $r>0$ and $\epsilon>0$ there is $N$ such that
$\overline{\mu_{n}}(M_{\geq r})\geq 1-\epsilon$ for every $n\geq N$, where
$M_{\geq r}$ denotes the $r$-thick part of $M$ and $\nu^{(i)}$ the $i$’th
convolution of $\nu$.

######  Remark 1.4.

$(i)$ It is possible to deduce the celebrated normal subgroup theorem of
Margulis by comparing Theorem 1.3 for $\Lambda\backslash G$ with the result of
Eskin and Margulis [EM02] about random walks on $\Gamma\backslash G$ where
$\Gamma\leq G$ is a lattice and $\Lambda\lhd\Gamma$ is a normal subgroup.
However, our proof relies on the Stück–Zimmer theorem (or rather a variant of
this theorem for stationary measures, see Theorem 1.9) which relies on the
intermediate factor theorem of Nevo and Zimmer [NZ02b]. Thus, as in Margulis’
original proof of the classical normal subgroup theorem, this approach also
traces back to the factor theorem of Margulis [M78].

$(ii)$ It follows from Theorem 1.1 that for higher rank manifolds, finite
volume is equivalent to bounded injectivity radius. It might be interesting to
obtain a quantitative version of that statement. It is also possible to deduce
the result of the seven authors [7s17, Theorem 1.5] from Theorem 1.3 by a
straightforward compactness argument.

###  1.2. Uniformly slim subgroups of semisimple groups

More generally, let $G$ be a connected centre-free semisimple Lie group. We
shall say that a discrete subgroup $\Lambda\leq G$ is uniformly slim if there
is a compact subset $C\subset G$ such that $\Lambda^{g}\cap
C\setminus\\{1\\}\neq\emptyset$ for every $g\in G$. In other words, $\Lambda$
is uniformly slim if and only if the locally symmetric space
$\Lambda\backslash G/K$ has bounded injectivity radius. Theorem 1.1 is a
special case of the following result (see Theorem 9.13 for a more general
statement where rank one factors with Kazhdan’s property (T) are allowed):

######  Theorem 1.5.

Suppose that all the simple factors of $G$ are of real rank at least $2$. A
discrete subgroup $\Lambda\leq G$ is uniformly slim if and only if there is a
nontrivial normal subgroup $H\lhd G$ such that $\Lambda\cap H$ is a lattice in
$H$.

We shall say that a subgroup $\Delta\leq\Gamma$ is a conjugate limit of
$\Lambda$ if $\Delta$ belongs to the closure of the conjugacy class
$\overline{\Lambda^{G}}\subset\text{Sub}(G)$ in the Chabauty topology (see
§2). It is obvious that a conjugate limit of a uniformly slim group is also
uniformly slim.

###  1.3. Random walk on $G/\Lambda$

Let $G$ be a connected semisimple Lie group. Let $\Lambda\subset G$ be a
discrete subgroup of $G$. Consider the sequence of probability measures

$\mu_{n}:=\frac{1}{n}\sum_{i=0}^{n-1}\int_{G}\delta_{\Lambda^{g}}d\nu^{(n)}(g).$

Every weak-* limit of the sequence $\mu_{n}$ is a $\nu$-stationary measure
supported on the closure of the orbit of $\Lambda$ in $\text{Sub}(G)$. We
would like to know if stationary random subgroups constructed in this way
retain some properties of $\Lambda$. An essential result of this paper is that
a stationary random subgroup is almost surely discrete (see also Theorem 2.2
for a general statement).

######  Theorem 1.6.

Let $\mu_{\infty}$ be a weak-* limit of $(\mu_{n})$. Then
$\mu_{\infty}(\text{Sub}_{d}(G))=1$ where $\text{Sub}_{d}(G)$ is the Chabauty
open set of discrete subgroups of $G$.

###  1.4. Stiffness and the Stück–Zimmer theorem for stationary measures

In view of Theorem 1.6 we are led to study stationary measures supported on
the space $\text{Sub}_{d}(G)$ of discrete subgroups of $G$, that is,
$\nu$-stationary measures $\mu$ with $\mu(\text{Sub}_{d}(G))=1$. Relying on
the remarkable theorems of Nevo and Zimmer [NZ02, NZ99] we establish that
every discrete stationary random subgroup of a higher rank group is (under a
certain irreducibility assumption with respect to the rank one factors) an
invariant random subgroup.

######  Theorem 1.7.

Let $G$ be a connected centre-free semisimple Lie group without compact
factors and real rank at least two. Let $\mu$ be a $\nu$-stationary measure on
$\text{Sub}_{d}(G)$. Suppose that $\mu$-almost every random subgroup
intersects trivially every rank one factor of $G$. Then $\mu$ is invariant.

Theorem 1.7 is a consequence of a decomposition theorem for stationary
measures (Theorem 6.5). Observe that in view of Theorem 6.5, the condition
that the intersection with every rank one factor is trivial implies that the
projection to every such factor is either non-discrete or trivial. The analog
of Theorem 1.7 does not hold for rank one groups (see Example 6.8), yet we
prove a week variant of the Nevo–Zimmer factor theorem for rank one groups
(see Theorem 5.1).

######  Remark 1.8.

Theorem 1.7 specialized to irreducible lattices in $G$ can also be deduced
from the recent results [BH20, BBHP20, C20].

For semisimple Lie groups with Kazhdan’s property (T) we deduce the following
generalization of the Stück–Zimmer theorem for discrete stationary random
subgroups:

######  Theorem 1.9.

Let $G$ be a connected centre-free semisimple Lie group without compact
factors. Suppose that $G$ has real rank at least $2$ and Kazhdan’s property
(T). Let $\mu$ be an ergodic $\nu$-stationary measure on $\text{Sub}_{d}(G)$.
Suppose that $\mu$-almost every random subgroup intersects trivially every
rank one factor of $G$. Then there is a semisimple factor $H\lhd G$ and a
lattice $\Gamma\leq H$ such that $\mu=\mu_{\Gamma}$.

Here $\mu_{\Gamma}$ denotes the invariant random subgroup obtained by the
pushforward of the probability measure from $H/\Gamma$ to
$\text{Sub}(H)\subset\text{Sub}(G)$ via the map $h\Gamma\mapsto h\Gamma
h^{-1}$.

###  1.5. The conclusion

Combining Theorem 1.6, Theorem 1.9 and local rigidity we deduce the following:

######  Theorem 1.10.

Let $G$ be a connected centre-free semisimple Lie group without compact
factors. Suppose that $G$ has real rank at least $2$ and Kazhdan’s property
(T). Let $\Lambda\leq G$ be a discrete subgroup. Suppose that for every
nontrivial semisimple factor $H\lhd G$ the intersection $\Lambda\cap H$ is not
a lattice in $H$. Suppose also that no discrete conjugate limit of $\Lambda$
intersects a rank one factor of $G$ in a Zariski dense subgroup. Then
$\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(i)}*\delta_{\Lambda}$ weakly converges
to $\delta_{\\{1\\}}$.

See §9 below for more general results of this nature. In particular Theorem
9.2 deals with a general discrete subgroup $\Lambda$ of a general semisimple
group $G$, not even assuming property (T). Loosely speaking, if $\Lambda$ does
not contain a lattice of a higher rank semisimple factor then every stationary
limit is supported on discrete subgroups of the product of rank one factors of
$G$.

######  Remark 1.11.

The main results of this work holds also for analytic groups over non-
archimedean local fields. The same proofs can be carried out in that
generality with minor adaptations. However since the Nevo–Zimmer factor
theorem [NZ02, NZ99] is written for real Lie groups, we decided to restrict to
that case as well. We remark that the proof of the Nevo–Zimmer theorem also
applies with minor changes to the non-archimedean setup.

Acknowledgment. We thank Uri Bader for sharing with us some insights
concerning stationary measures and Poisson boundaries. Our work was supported
by the National Science Foundation under Grant No. DMS-1928930 while the
authors participated in a program hosted by the Mathematical Sciences Research
Institute in Berkeley, California, during the Fall 2020 semester. The second
author was partially supported by the Israel Science Foundation grant No.
2919/19.

##  2\. Random walks on the space of discrete subgroups

Let $G$ be a connected centre-free semisimple Lie group without compact
factors and let $K$ be a maximal compact subgroup.

###  2.1. The measure associated to $G$

We let $\nu_{G}$ be the probability measure

$\nu_{G}=\eta_{K}*\delta_{s}*\eta_{K}$

defined in [GLM, §8]. Here $\eta_{K}$ denotes the normalized Haar measure on
$K$ and $s\in G$ is a certain regular semisimple element with sufficiently
good expanding properties when acting on the unipotent radical of a fixed
minimal parabolic subgroup (see [GLM, §6]). We will refer to $\nu_{G}$ as the
probability measure associated to $G$. Note that if $G$ is semisimple,
$G=G_{1}\times\ldots\times G_{n}$, we have
$\nu_{G}=\nu_{G_{1}}*\cdots*\nu_{G_{n}}$ and the measures $\nu_{G_{i}}$
pairwise commute.

###  2.2. The discreteness radius

Fix a norm $\|\cdot\|$ on the Lie algebra $\text{Lie}(G)$ such that
$\text{exp}:\text{Lie}(G)\to G$ restricted to the unit ball
$B(1)=\\{X\in\text{Lie}(G):\|X\|<1\\}$ is a well defined diffeomorphism. For
$r\leq 1$ denote $B(r)=\\{X\in\text{Lie}(G):\|X\|<r\\}$. For a discrete group
$\Lambda\subset G$ set

$\mathcal{I}(\Lambda)=\sup\\{r\leq 1:\exp B(r)\cap\Gamma=\\{1\\}\\}.$

We call $\mathcal{I}(\Gamma)$ the discreteness radius of $\Lambda$.

###  2.3. The Margulis function on the space of discrete subgroups of $G$

We denote by $\text{Sub}(G)$ the space of closed subgroups of $G$ equipped
with the Chabauty topology and by $\text{Sub}_{d}(G)$ the subset of discrete
subgroups of $G$. Since $G$ has no small subgroups, $\text{Sub}_{d}(G)$ is
open (see [G18, Lemma 1.1]). We will say that a measure $\mu$ on
$\text{Sub}(G)$ is supported on the set of discrete subgroups if
$\mu(\text{Sub}_{d}(G))=1$. A stationary measure supported on
$\text{Sub}_{d}(G)$ will be called a discrete stationary random subgroup.

An essential result established in [GLM] is that there is a positive constant
$\delta=\delta(G)$ such that

$u(\Gamma):=\mathcal{I}(\Gamma)^{-\delta}$

is a Margulis function on $\text{Sub}_{d}(G)$ with respect to $\nu_{G}$.

######  Theorem 2.1 ([GLM], Theorem 1.4).

There exist $0<c<1,b\geq 0$ such that, for every discrete subgroup $\Gamma\leq
G$,

(1) $\int_{G}u(\Gamma^{g})d\nu_{G}(g)\leq cu(\Gamma)+b.$

We remark that the constants $\delta,c$ and $b$ are constructed explicitly in
[GLM] in order to prove certain effective results and in particular a
quantitative version of the Kazhdan–Margulis theorem.

###  2.4. Stationary limit are discrete

The main result of this section is that any stationary limit of a measure
supported on discrete subgroups of $G$ is almost surely discrete. This is a
key ingredient in the proofs of Theorem 1.1 and Theorem 1.3, as well as the
results of §9.

######  Theorem 2.2.

Let $\mu$ be a probability measure on $\text{Sub}_{d}(G)$. Let

$\mu_{n}=\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(n)}*\mu$

and let $\mu_{\infty}$ be a weak-* limit of $\mu_{n}$. Then $\mu_{\infty}$ is
supported on the set of discrete subgroups of $G$, that is, $\mu_{\infty}$ is
a discrete stationary random subgroup.

The proof is reminiscent of [EM02].

###### Proof.

By restricting to compact subsets of $\text{Sub}_{d}(G)$, we may allow ourself
to suppose that $\mu$ is compactly supported and that $A:=\int
u(\Lambda)d\mu(\Lambda)$ is finite.

To prove Theorem 2.2 we need to show that

(2) $\lim_{\epsilon\to
0}\mu_{\infty}({\\{\Lambda:\mathcal{I}(\Lambda)<\epsilon\\}})=0.$

Inequality (1) implies that

$\int u(\Lambda)d(\nu_{G}*\mu)(\Lambda)=\int
u(\Lambda^{g})d\nu_{G}(g)d\mu(\Lambda)\leq cA+b.$

Furthermore, iterating Condition (1) and summing the resulting geometric
series we get

$\int u(\Lambda)d\mu_{n}(\Lambda)=\int
u(\Lambda)d(\nu_{G}^{(n)}*\mu)(\Lambda)<c^{n}A+C,$

with $C:=b/(1-c)$, uniformly for all $n$. Set $M=A+C$. Then, for every
$\tau>0$ and $n\geq 1$ we have

$\mu_{n}(\\{\Lambda:u(\Lambda)\geq M\tau^{-1}\\})<\tau.$

Setting $\epsilon=(\tau/M)^{\frac{1}{\delta}}$ we get

$\mu_{n}(\\{\Lambda:\mathcal{I}(\Lambda)\leq\epsilon\\})<\tau.$

Taking $n\to\infty$ gives
$\mu_{\infty}(\\{\Lambda:\mathcal{I}(\Lambda)\leq\epsilon\\})<\tau$, and
letting $\tau\to 0$ we get (2). ∎

##  3\. Essential results about discrete stationary random subgroups

In this section we assemble some results about discrete stationary random
subgroups that will be essential in the proofs of the main results. The
measure $\nu_{G}$ introduced in §2 is bi-$K$-invariant. In view of the Iwasawa
decomposition, $K$ acts transitively on $G/P$ where $P\leq G$ is a minimal
parabolic subgroup. Therefore there is a unique Borel regular $K$-invariant
probability measure $\mu_{P}$ on $G/P$, and the space $(G/P,\mu_{P})$ is the
Poisson boundary for $(G,\nu)$ whenever $\nu$ is a bi-$K$-invariant
probability measure on $G$.

######  Remark 3.1.

From the universal property of the Poisson boundary we see that if
$\nu_{1},\nu_{2}$ are two probability measures on $G$ which correspond to the
same Poisson boundary and Poisson measure $(B,\mu)$, then they share the same
stationary measures on every $G$-space $X$. Indeed, a measure $\mu_{0}$ on $X$
is stationary with respect to $\nu_{1}$ (equivalently $\nu_{2}$) if and only
if it is the $\mu$-barycentre for some measurable map from $B$ to the space
$\text{Prob}(X)$ of probability measures on $X$.

The measure $\nu_{G}$ is not smooth111It is possible to show that some finite
power $\nu_{G}^{(n)}$ is smooth, but in view of Remark 3.1 we will not need
that. but obviously $G$ admits some smooth bi-$K$-invariant probability
measure. Therefore, in view of Remark 3.1 statements about stationary measures
with respect to a measure on $G$ which is assumed to be smooth will remain
valid also for the non-smooth measure $\nu_{G}$. For simplicity we will
consider, without repeating it, only bi-$K$-invariant probability measures
$\nu$ on $G$ although most of the statements apply to general (smooth)
measures. This will allow us to make use of the special properties of
$\nu_{G}$ and in particular properties that follow from Inequality (1) and
from Theorem 2.2. Thus in the sequel $\nu$ will refer to an arbitrary
bi-$K$-invariant probability measure on $G$ while $\nu_{G}$ is the specific
measure given in §2.1.

Let us denote by $\text{d-SRS}(G)$ the space of all discrete $\nu$-stationary
random subgroups of $G$. The following result from [GLM] is a consequence of
Inequality (1):

######  Theorem 3.2 ([GLM], Theorem 1.2).

The space $\text{d-SRS}(G)$ is weakly uniformly discrete. That is, for every
$\epsilon>0$ there is an identity neighbourhood $U\subset G$ such that for
every $\mu\in\text{d-SRS}(G)$,

$\mu(\Lambda:\Lambda\cap U\neq\\{1\\})\leq\epsilon.$

Let us recall the straightforward proof when $\mu$ is a compactly supported
discrete stationary random subgroup. In that case Inequality (1) gives:

$\int u(\Lambda)d\mu=\int u(\Lambda)d\nu*\mu\leq c\int u(\Lambda)d\mu+b.$

It follows that $\int u(\Lambda)d\mu\leq C:=b/(1-c)$. Thus
$\mu(\\{\Lambda:u(\Lambda)\geq C/\epsilon\\})\leq\epsilon$. Thus we can take
$U$ to be the ball of radius $(\epsilon/C)^{\frac{1}{\delta}}$ around $1_{G}$.

######  Corollary 3.3 ([GLM], Corollary 1.5).

The space $\text{d-SRS}(G)$ is weak-* compact.

###### Proof.

It is obvious that a limit of stationary measures is stationary and it follows
from Lemma 3.2 that a limit of discrete stationary random subgroups is also
discrete. ∎

We deduce from [BSh06, §2] that the extreme points of the compact convex space
$\text{d-SRS}(G)$ are ergodic (see in particular [BSh06, Corollary 2.7]). Thus
by the Choquet integral theorem every discrete stationary random subgroup is a
barycentre of some probability measure on the set of ergodic discrete
stationary random subgroups. This fact allows us to assume ergodicity when
proving various results about discrete stationary random subgroups.

Let $P$ be a minimal parabolic subgroup of $G$ with Langlands decomposition
$P=MAN$.

######  Lemma 3.4.

Let $H\subset G$ be an algebraic subgroup. The homogeneous space $G/H$ admits
a $\nu$-stationary probability measure if and only if $H$ contains $AN$, up to
conjugacy.

After writing the proof of Lemma 3.4 we found out that the same statement was
already proven in [NZ02, Prop 3.2]. Our proof is different and self-contained
so we present it below for completeness.

###### Proof.

If $H$ contains a conjugate of $AN$ then $G/H$ is compact so the existence of
a $\nu$-stationary probability measure is clear. Now assume that $\mu$ is a
$\nu$-stationary probability measure on $G/H$. Since $(G/P,\mu_{P})$ is the
Poisson boundary we get a $G$-equivariant map $\kappa\colon
G/P\to\mathcal{P}(G/H)$ such that $\mu=\int_{G/P}\kappa(gP)d\mu_{P}(gP).$ The
measures $\kappa(gP)$ are $gPg^{-1}$-invariant probability measures almost
surely. By conjugating $P$ if necessary we can assume without loss of
generality that $\kappa(P)$ is a $P$-invariant probability measure. We will
show that the existence of such a measure implies that $H$ contains a
conjugate of $AN$. Let $\mu_{0}$ be an ergodic $AN$-invariant component of
$\mu$. By Chevalley’s theorem [Bor, 5.1] there is a rational representation
$G\curvearrowright V$ and a line $[v]\in\mathbb{P}^{1}(V)$ such that ${\rm
Stab}_{G}[v]=H$. We will think of $\mu_{0}$ as an ergodic $AN$-invariant
probability measure on $\mathbb{P}^{1}(V)$ supported on the $G$ orbit of
$[v]$. By Lemma 3.5, we must have $\mu_{0}=\delta_{[gv]}$ for some $g\in G$
and $[gv]$ must be fixed by $AN$. If follows that $gHg^{-1}\supset AN$. ∎

The following Lemma 3.5 is required also in the proof of Lemma 5.2.

######  Lemma 3.5.

Let $V$ be a rational real representation of $AN$. Then, any ergodic
$AN$-invariant probability measure on $\mathbb{P}^{1}(V)$ is supported on a
single line fixed by $AN$.

###### Proof.

Let $\mu$ be an ergodic $AN$-invariant probability measure on
$\mathbb{P}^{1}(V)$. The group $N$ is a unipotent algebraic group so any
rational representation of $N$ has a fixed vector [Bor, 4.8]. Let $V^{\prime}$
be the subspace of $N$-fixed vectors. It is preserved by $A$. Since $A$ is an
$\mathbb{R}$-split torus, any rational representation of $A$ over $\mathbb{R}$
decomposes into a direct sum of $A$-eigenspaces [Bor, 8.4]. We deduce that
there exists a one-dimensional subspace $V_{1}\subset V^{\prime}$ which is
preserved by $AN$. Reasoning inductively we construct a basis
$e_{1},\ldots,e_{d}$ of $V$ such that
$V_{i}=\mathbb{R}e_{1}+\ldots+\mathbb{R}e_{i}$ are preserved by $AN$ and each
$e_{i}$ is an eigenvector of $A$. Write $\chi_{i}$ for the character of $A$
such that $ae_{i}=\chi_{i}(a)e_{i}.$

Let $i_{0}$ be the minimal index for which $\mu(\mathbb{P}^{1}(V_{i_{0}}))>0$.
The sets $\mathbb{P}^{1}(V_{i_{0}})$ are all $AN$-invariant, so by ergodicity
$\mu(\mathbb{P}^{1}(V_{i_{0}})\setminus\mathbb{P}^{1}(V_{i_{0}-1}))=1.$
Consider the map
$\iota\colon\mathbb{P}^{1}(V_{i_{0}})\setminus\mathbb{P}^{1}(V_{i_{0}-1})\to
V_{i_{0}}$

$\iota([x_{1},\ldots,x_{i_{0}-1},x_{i_{0}}]):=(x_{1}/x_{i_{0}},\ldots,x_{i_{0}-1}/x_{i_{0}},1).$

Let $W$ be the space $V_{i_{0}}$ with the action of $AN$ given by
$anw=\chi_{i}(a)^{-1}anw.$ The map
$\mathbb{P}^{1}(V_{i_{0}})\setminus\mathbb{P}^{1}(V_{i_{0}-1})\to W$ is
$AN$-equivariant. The measure $\iota_{*}\mu$ is an ergodic $AN$-invariant
probability measure on $W$. By Lemma 3.6, $\iota_{*}\mu=\delta_{w}$ for some
$AN$ invariant vector $w\in W$. This means that $\mu$ itself was supported on
the line $[w+e_{i_{0}}]$, which is fixed by $AN$. ∎

######  Lemma 3.6.

Let $W$ be a finite dimensional rational representation of $AN$. Any ergodic
$AN$-invariant probability measure on $W$ is of the form $\delta_{w}$ for an
$AN$-fixed vector $w\in W$.

###### Proof.

We prove the statement by induction, with cases $\dim W=1,2$ serving as the
induction base. If $\dim W=1$ then $N$ acts trivially because unipotent
actions must fix a non-zero vector. If $A$ acts non-trivially then
$\mu=\delta_{0}$ because no probability measure on
$\mathbb{R}\setminus\\{0\\}$ can be invariant under dilations. Otherwise, $AN$
acts trivially and $\mu=\delta_{w}$ for some $w\in W$, by ergodicity.

Suppose $\dim W=2$. Let $e_{1},e_{2}$ be a basis of $W$ such that $e_{1}$ is
$N$-invariant and both $e_{1},e_{2}$ are eigenvectors of $A$. If the action of
$N$ is nontrivial, then there exists $n_{0}\in N$ and
$x_{0}\in\mathbb{R}\setminus\\{0\\}$ such that
$n_{0}(e_{2})=e_{2}+x_{0}e_{1}$. For every compact set $K\subset W$ the
intersection $\bigcap_{k\in\mathbb{Z}}n_{0}^{k}K\subset
K\cap(\mathbb{R}e_{1}),$ so every $N$-invariant probability measure on $W$
must have $\mu(\mathbb{R}e_{1})=1$ and the lemma follows now from the one
dimensional case. Consider the case that the action of $N$ is trivial. There
are rational characters $\chi_{1},\chi_{2}$ of $A$ such that
$ae_{i}=\chi_{i}(a)e_{i}$. Let $W^{\prime}=\sum_{\chi_{i}=1}\mathbb{R}e_{i}$.
Then for any compact subset $K\subset W$ we have $\bigcap_{a\in A}aK\subset
K\cap W^{\prime}$. It follows that $\mu$ is supported on $W^{\prime}$. The
action of $AN$ on $W^{\prime}$ is trivial so by ergodicity $\mu=\delta_{w}$
for some $w\in W^{\prime}$.

We move to the general case $\dim W\geq 3$. Let $W_{1}$ be a one dimensional
subspace preserved by $AN$. By the inductive hypothesis, the pushforward of
$\mu$ to $W/W_{1}$ is supported on a single element. It follows that $\mu$ is
supported on a single line $w+W_{1}$ for some $w\in W$. Hence
$\mu(W_{1}+\mathbb{R}w)=1$. We have $\dim(W_{1}+\mathbb{R}w)\leq 2$ so the
lemma follows from the first two cases. ∎

######  Lemma 3.7.

Let $\Phi$ be a simple root system spanning a Euclidean space $V$. Let
$V_{0}\subset V$ be a proper subspace and let $S=\Phi\cap V_{0}$. Then

  1. (1)

$\Phi\setminus S$ generates the whole root system $\Phi$.

  2. (2)

$\Phi\setminus S$ spans $V$.

###### Proof.

For any subset $F\subset\Phi$, let $\langle F\rangle$ be the smallest subset
of $\Phi$ containing $F$ that is closed under taking reflections. For the
first assertion we need to show $\langle\Phi\setminus S\rangle=\Phi$. Let
$S_{0}=S,E_{0}=V_{0}$. We define inductively the subsets $S_{i}$ and subspaces
$E_{i}$.

$E_{i+1}=E_{i}\cap(\Phi\setminus S_{i})^{\perp},\quad S_{i+1}=\Phi\cap
E_{i+1}.$

By construction $S_{i+1}\subset S_{i}$ and $E_{i+1}\subset E_{i}$. We argue
that $\langle\Phi\setminus S_{i}\rangle=\langle\Phi\setminus S_{i+1}\rangle.$
The inclusion $\langle\Phi\setminus S_{i}\rangle\subset\langle\Phi\setminus
S_{i+1}\rangle$ is clear, so it is enough to show that $S_{i}\setminus
S_{i+1}\subset\langle\Phi\setminus S_{i}\rangle.$ Let $\lambda\in
S_{i}\setminus S_{i+1}$. By definition, we must have $\lambda\not\in E_{i+1}$,
so there exists a root $\alpha\not\in S_{i}$ such that
$\langle\lambda,\alpha\rangle\neq 0$. The reflection

$s_{\alpha}(\lambda)=\lambda-\frac{2\langle\lambda,\alpha\rangle}{\|\alpha\|^{2}}\alpha$

is not in $S_{i}$ because $\lambda\in E_{i}$ and $\alpha\not\in E_{i}$. We
deduce that $\lambda=s_{\alpha}(s_{\alpha}(\lambda))\in\langle\Phi\setminus
S_{i}\rangle.$ This proves that $\langle\Phi\setminus
S_{i}\rangle=\langle\Phi\setminus S_{i+1}\rangle.$

The sequence of sets $S_{i}$ eventually stabilizes, so we have $S_{i+1}=S_{i}$
for some $i$. Therefore $S_{i}\subset(\Phi\setminus S_{i})^{\perp}.$ The root
system $\Phi$ is simple, so this is possible only if $S_{i}=\emptyset.$ We
deduce that $\langle\Phi\setminus S_{0}\rangle=\langle\Phi\setminus
S_{1}\rangle=\ldots=\langle\Phi\setminus S_{i}\rangle=\Phi.$ The second
assertion trivially follows from the first. ∎

######  Remark 3.8.

The assumption that $S$ is contained in a proper subspace is necessary in
Lemma 5.2. For example, the root systems $B_{2},C_{2},G_{2},B_{3},C_{3}$
decompose as unions of two proper root subsystems.

######  Lemma 3.9.

The cenralizer of $AN$ in $G$ is trivial.

###### Proof.

The lemma easily reduces to the case where $G$ is simple, so let us assume
that from now on. The centralizer of $A$ is $MA$. It follows that $C_{G}(AN)$
is a normal subgroup of the reductive group $MA$. Therefore it is enough to
show that every semisimple element in $C_{G}(AN)$ is trivial. Let $\gamma\in
C_{G}(AN)$ be a semisimple element. Let $T$ be a maximal torus of $G$
containing $\gamma$. Let $\Phi$ be the set of roots of $T$ in
$\mathfrak{g}_{\mathbb{C}}$. The roots naturally lie in the Euclidean space
$X^{*}(T)\otimes\mathbb{R}$ and form a root system in the classical sense. Let
$E\subset X^{*}(T)\otimes\mathbb{R}$ be the proper subspace spanned by the
roots in $\mathfrak{m}_{\mathbb{C}}$. Then, $\Phi\setminus E$ corresponds to
the roots in $\mathfrak{n}$ and their opposites. Since $\gamma$ commutes with
$N$, we must have $\xi(\gamma)=1$ for every root $\xi\in\Phi\setminus E$. On
the other hand, by Lemma 3.7, the roots in $\Phi\setminus E$ generate $\Phi$,
so $\xi(\gamma)=1$ for every root in $\Phi$. This means that ${\rm
Ad}(\gamma)$ acts trivially on $\mathfrak{g}_{\mathbb{C}}$. Since $G$ is
centre-free, $\gamma=1$. ∎

Let $2^{G}$ denote the compact space of closed subsets of $G$ equipped with
the Chabauty topology and consider the $G$ action by conjugation. Let
$\mathcal{F}(G)\subset 2^{G}$ be the set of finite subsets of $G$.

######  Lemma 3.10.

For any finite set $F\subset G\setminus\\{1\\}$, the $\nu_{G}$-stationary
measure

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(n)}*\delta_{F}$

is the Dirac measure on the empty set.

###### Proof.

It is enough to prove the assertion for $F=\\{\gamma\\},\gamma\in
G\setminus\\{1\\}$. Moreover, by projecting to a factor we may suppose that
$G$ is simple. Let $\mu$ be the limit in question. If it is not the Dirac
delta on the empty set then it gives us a $\nu$-stationary probability measure
on the closure of the conjugacy class $\overline{\gamma^{G}}$. Since
$\overline{\gamma^{G}}$ is a finite union of conjugacy classes, by restricting
to one, say $\gamma_{0}^{G}$, we obtain a finite positive $\nu$-stationary
measure on $G/G_{\gamma_{0}}$. By Lemma 3.4, up to replacing $\gamma_{0}$ by a
conjugate, $G_{\gamma_{0}}$ contains $AN$. By Lemma 3.9 we get $\gamma_{0}=1$.
Thus, we are left with showing that $\mu$ cannot have an atom at $\\{1\\}$.
Now if $\mu(\\{1\\})=\epsilon>0$ then $1$ is a conjugate limit of $\gamma$ and
$\gamma$ is unipotent. In that case by applying Theorem 2.2 to the discrete
cyclic group $\langle\gamma\rangle$ we can choose an identity neighborhood $U$
such that
$\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(n)}*\delta_{\\{\gamma\\}}(U)<\epsilon$,
a contradiction. ∎

######  Corollary 3.11.

The only ergodic $\nu$-stationary probability measures on $\mathcal{F}(G)$ are
$\delta_{\\{1\\}}$ and $\delta_{\emptyset}$.

###### Proof.

In view of the dominated convergence theorem it follows from Lemma 3.10 that
the only $\nu$-stationary probability measure on $G$ with respect to the
action by conjugation is $\delta_{\\{1\\}}$. Since the average of the Dirac
measures on a finite set which is chosen randomly with respect to a
$\nu$-stationary measure on $\mathcal{F}(G)$ produces such a measure, the
corollary follows. ∎

We will also make use of the following beautiful result:

######  Proposition 3.12 (Bader–Shalom [BSh06]).

Let $H=H_{1}\times H_{2}$, let $\nu_{i}$ be a probability measure on $H_{i}$,
$i=1,2$ and let $\nu=\nu_{1}*\nu_{2}$. Let $X$ be an $H$-space and $\mu$ a
$\nu$-stationary measure on $X$. Then $\mu$ is $\nu_{i}$-stationary.

###### Proof.

In view of [BSh06, Corollary 2.7] it is enough to prove the result when $\mu$
is ergodic. For $\mu$ ergodic this is the statement of [BSh06, Lemma 3.1]. ∎

######  Lemma 3.13.

Let $H=H_{1}\times H_{2}$ be a product of centre-free semisimple Lie groups
without compact factors and let $\mu$ be a $\nu_{H}$-stationary random
discrete subgroup. If the projection to $H_{1}$ is almost surely discrete and
the intersection with $H_{2}$ is almost surely trivial, then $\mu$ is
supported on subgroups of $H_{1}$.

###### Proof.

Let us denote by $\pi_{i}$ the projection to $H_{i}$ for $i=1,2$. Suppose by
way of contradiction that $\mu$ is not supported on subgroups of $H_{1}$. Then
for any sufficiently large ball $B$ in $G_{1}$ we have

$\mu\\{\Lambda:\exists\alpha\in\Lambda~{}\text{with}~{}\pi_{1}(\alpha)\in
B,\pi_{2}(\alpha)\neq 1\\}>0.$

Considering the map $\Lambda\mapsto\pi_{2}(\Lambda\cap\pi_{1}^{-1}(B))$, the
measure $\mu$ induces a $\nu_{H_{2}}$-stationary finite positive measure on
$\mathcal{F}(H_{2})$ which is not a combination of $\delta_{\\{1\\}}$ and
$\delta_{\emptyset}$. This contradicts Corollary 3.11. ∎

######  Remark 3.14.

Given a locally compact group $G$ and a probability measure $\mu$ on
$\text{Sub}_{d}(G)$ it is possible to construct a probability $G$-space
$(X,m)$ such that $\mu$ is the pushforward of $m$ via the stabilizer map
$X\to\text{Sub}(G),~{}x\mapsto G_{x}$. This is proven in [7s17, Theorem 2.6]
under the assumption that $\mu$ is an IRS, but the proof applies to any
$\mu\in\text{Prob}(\text{Sub}(G))$ which is supported on unimodular subgroups.
Thus, the study of probability measures on $\text{Sub}_{d}(G)$ is equivalent
to the study of discrete stabilizers of probability $G$-spaces. We will not
make use of this fact.

##  4\. A decomposition result for Invariant Random Subgroups

Let $G$ be a connected centre-free semisimple Lie group without compact
factors. It follows from the Borel density theorem for IRS [7s17, GL18] that
for every ergodic discrete invariant random subgroup $\mu$ on $G$ there is a
decomposition $G=G^{\prime}\times G^{\prime\prime}$ such that the projection
to $G^{\prime}$ is discrete and Zariski dense and the projection to
$G^{\prime\prime}$ is trivial almost surely. The following results (see
Theorem 4.1 and Corollary 4.4 below) generalize to invariant random subgroups
the classical decomposition-to-irreducible-factors theorem for lattices in
semisimple groups (see [Rag89, Theorem 5.22]):

######  Theorem 4.1.

Let $G=G_{1}\times\cdots\times G_{n}$ be a connected centre-free semisimple
Lie group without compact factors and with simple factors
$G_{i},~{}i=1,\ldots,n$. Let $\mu$ be an ergodic discrete invariant subgroup
in $G$. Then $G$ decomposes to a product of semisimple factors
$G=H_{1}\times\ldots\times H_{k}$ with $1\leq k\leq n$, such that almost
surely the projection of a random subgroup to each $H_{i}$ is discrete while
the projection to each proper factor of $H_{i}$ is dense.

The proof relies on the following:

######  Lemma 4.2.

Let $H=H_{1}\times H_{2}$ be a product of centre-free semisimple Lie groups
without compact factors. Let $\mu$ be an ergodic discrete IRS in $H$ which
projects discretely to $H_{1}$ and Zariski densely to $H_{2}$. Then the
intersection of a random subgroup with $H_{2}$ is nontrivial almost surely.

###### Proof of Lemma 4.2.

The lemma follows immediately from Lemma 3.13. ∎

######  Remark 4.3.

This lemma could also be proved directly (without referring to the special
properties of the measure $\nu_{H}$) by applying disintegration of measures
with respect to the factor map
$(\text{Sub}_{H},\mu)\to(\text{Sub}_{H_{1}},\pi_{*}\mu)$. Indeed, if the
intersection with $H_{2}$ is trivial almost surely, then since the projection
to $H_{2}$ is Zariski dense, the $H_{2}$ action on a generic fibre is free. By
constructing a measurable section for the $H_{2}$ action on a fibre one may
pull the fibre measure to a left $H_{2}$-invariant probability measure on
$H_{2}$. This is absurd since $H_{2}$ is not compact.

###### Proof of Theorem 4.1.

Let $G=G_{1}\times\cdots\times G_{n}$ and $\mu$ be as in the statement of the
theorem. We may suppose that a random subgroup is almost surely Zariski dense.
If the projection of a random subgroup to every proper semisimple factor of
$G$ is dense almost surely then $\mu$ is irreducible. Otherwise there is a
proper decomposition $G=H\times H^{\prime}$ such that the projection to $H$ is
almost surely discrete. We claim that the projection to $H^{\prime}$ is also
discrete and hence we can deduce the result by induction on the number of
simple factors. Suppose by way of contradiction that the projection to
$H^{\prime}$ is non-discrete and let $H_{2}\lhd H^{\prime}$ be the connected
component of its closure. Then $H^{\prime}$ decomposes as
$H^{\prime}=H^{\prime\prime}\times H_{2}$ with $H_{2}$ nontrivial. It follows
that $\mu$ projects discretely to $H_{1}:=H\times H^{\prime\prime}$ and
densely to $H_{2}$. Since the intersection of every subgroup of $G$ with
$H_{2}$ is normalized by the projection of that subgroup to $H_{2}$, a
discrete subgroup of $G$ that projects densely to $H_{2}$ must intersect it
trivially. Therefore a $\mu$-random subgroup intersects $H_{2}$ trivially
almost surely. A contradiction to Lemma 4.2. ∎

It follows from Theorem 4.1 that an ergodic discrete IRS $\mu$ in
$G=G_{1}\times\cdots\times G_{n}$ is associated with two other IRS
$\tilde{\mu}$ and $\overline{\mu}$ given (respectively) by its projection to
and the intersection with the semisimple factors $H_{j}$. Both $\tilde{\mu}$
and $\overline{\mu}$ are products,
$\tilde{\mu}=\prod\tilde{\mu}_{j},~{}\overline{\mu}=\prod\overline{\mu}_{j}$,
where $\tilde{\mu}_{j}$ and $\overline{\mu}_{j}$ are discrete IRS on $H_{j}$.
Obviously $\tilde{\mu}_{j}$ is irreducible. We claim that also
$\overline{\mu}_{j}$ is irreducible and nontrivial. If $\mu$ is irreducible
then $\mu=\tilde{\mu}=\overline{\mu}$ and there is nothing to prove.
Otherwise, it follows from Lemma 4.2 that $\overline{\mu}_{j}$ is nontrivial
for every $j$ for which $\tilde{\mu}_{j}$ is nontrivial. Fix
$j\in\\{1,\ldots,k\\}$. If $H_{j}$ is simple then there is nothing further to
prove. Suppose that $H_{j}$ is not simple. Let $F_{j}\lhd H_{j}$ be a proper
nontrivial semisimple factor of $H_{j}$. Let $\Delta$ be a $\mu$ random
subgroup and let $\tilde{\Delta}_{j}$ and $\overline{\Delta}_{j}$ be the
corresponding random subgroups in $H_{j}$, that is the projection and the
intersection with $H_{j}$. We need to show that $F_{j}\overline{\Delta}_{j}$
is dense in $H_{j}$. Note that $\overline{\Delta}_{j}\cap F_{j}$ must be
trivial, since it is both discrete and normalized by the projection of
$\tilde{\Delta}_{j}$ to $F_{j}$, which is dense. Consider the identity
connected component of the closure of $\overline{\Delta}_{j}F_{j}$ and denote
it by $N_{j}$. We need to show that $N_{j}=H_{j}$ and indeed, if $N_{j}$ is a
proper subgroup then again we deduce that $\overline{\Delta}_{j}$ is not
contained in $N_{j}$. However, by construction the projection of
$\overline{\Delta}_{j}$ to $H_{j}/N_{j}$ is discrete. This is absurd since
this nontrivial discrete group is normalized by the dense projection of
$\tilde{\Delta}_{j}$ to $H_{j}/N_{j}$. Thus we have established the following:

######  Corollary 4.4.

Let $G=G_{1}\times\cdots\times G_{n}$ be as in Theorem 4.1. Let $\mu$ be an
ergodic Zariski dense discrete invariant random subgroup in $G$. Let
$G=H_{1}\times\ldots\times H_{k}$ be the decomposition of $G$ to
$\mu$-irreducible factors. For every $j$ let $\overline{\mu}_{j}$ be the IRS
in $H_{j}$ obtain by the intersection of the $\mu$-random subgroup with
$H_{j}$. Then $\overline{\mu}_{j}$ is a non-trivial irreducible discrete IRS
of $H_{j}$.

##  5\. Stationary measures of rank one groups

The celebrated factor theorem of Nevo and Zimmer [NZ02] (Theorem 6.1) is
concerned with stationary measures of higher rank semisimple Lie groups and as
shown in [NZ99, Theorem B] the analog result is not true for simple Lie groups
of rank one. In this section we establish the following weak version of
Nevo–Zimmer factor theorem for rank one groups.

######  Theorem 5.1.

Let $G$ be a centre-free simple rank-one real Lie group with a smooth
probability measure $\nu$. Let $(X,\mu)$ be a non-trivial ergodic probability
$\nu$-stationary non-free $G$-system. Then either $(X,\nu)$ has discrete
Zariski dense stabilizers almost surely or there exists a $G$-equivariant map
$(X,\mu)\to(G/P,\mu_{P})$ where $P$ is a minimal parabolic subgroup of $G$.

######  Lemma 5.2.

Let $G$ be a simple real Lie group. Let $V$ be a real linear representation of
$G$ without fixed points. Fix a minimal parabolic $P\subset G$ with Langlands
decomposition $P=MAN$. Every ergodic $\nu$-stationary measure $\mu$ on the
projective space $\mathbb{P}^{1}(V)$ is supported on a single $G$-orbit $G[v]$
where the stabilizer of $[v]$ is a proper subgroup of $G$ that contains $AN$.

###### Proof.

Let $(G/P,\mu_{P})$ be the Poisson boundary. We get a $G$-equivariant map
$\kappa\colon G/P\to{\rm Prob}(\mathbb{P}^{1}(V))$ such that
$\int_{G/P}\kappa(gP)d\mu_{P}(gP)=\mu$. Each $\kappa(gP)$ is a
$gPg^{-1}$-invariant measure on $\mathbb{P}^{1}(V)$. By Lemma 3.5 $\kappa(gP)$
is supported on $gANg^{-1}$ fixed lines almost surely. It follows that almost
all points in the support of $\mu$ are stabilized by a conjugate of $AN$.
Since $G/AN$ is compact, the $G$-orbits of these points are closed in
$\mathbb{P}^{1}(V)$. Using ergodicity, we deduce that the measure $\mu$ must
be supported on a single $G$ orbit, with stabilizer conjugate to some closed
subgroup $H\supset AN$. If the group $G$ fixes a line then it must fix it
pointwise, because it is simple. We assumed that $V$ has no fixed subspaces,
so $H$ is a proper subgroup. ∎

######  Lemma 5.3.

Let $G$ be a connected simple rank one real Lie group. Let $P$ be a minimal
parabolic of $G$ with Langlands decomposition $P=MAN$. Let $H$ be closed
subgroup containing $AN$. Then either $H\subset P$ or $H=G$.

###### Proof.

Let $\mathfrak{h},\mathfrak{p},\mathfrak{m},\mathfrak{a},\mathfrak{n}$ be the
Lie algebras of $H,P,M,A,N$ respectively. Let $\Theta$ be a Cartan involution
of $G$ stabilizing $A$. Let $\Sigma$ be the root system of $A$ and let
$\Sigma^{+}$ be the set of positive roots. For any
$\lambda\in\mathfrak{h}^{*}$ let

$\mathfrak{g}_{\lambda}=\\{X\in\mathfrak{g}|[Y,X]=\lambda(Y)\textrm{ for
}Y\in\mathfrak{a}\\}.$

We have

$\mathfrak{p}=\mathfrak{g}_{0}+\sum_{\alpha\in\Phi^{+}}\mathfrak{g}_{\alpha},\qquad\mathfrak{n}=\sum_{\alpha\in\Phi^{+}}\mathfrak{g}_{\alpha},\qquad\mathfrak{n}^{-}=\sum_{\alpha\in\Phi^{+}}\mathfrak{g}_{-\alpha},\\\
$

where $\mathfrak{n}^{-}$ is the Lie algebra of the unipotent radical of the
opposite parabolic.

First consider the case $\mathfrak{h}\subset\mathfrak{p}$. Let $H_{0}$ be the
connected component of $H$. We have $H_{0}\subset P$ and we need to show
$H\subset P$. Let $h\in H$. The set of maximal split tori in $H_{0}$ forms a
single conjugacy class, so there exists an $h_{0}\in H_{0}$ such that
$h_{1}:=hh_{0}^{-1}$ normalizes $A$. The rank of $G$ is one so either $h_{1}$
commutes with $A$ or $h_{1}ah_{1}^{-1}=a^{-1}$ for every $a\in A$. In the
first case $h_{1}\in Z_{G}(A)=MA\subset P$. In the second
$h_{1}\mathfrak{n}h_{1}^{-1}=\mathfrak{n}^{-}$, which is impossible because
$h_{1}$ normalizes $\mathfrak{h}$. We deduce that $h_{1}\in P$ and
consequently $h\in P$. This proves $H\subset P$.

Now consider the case $\mathfrak{h}\not\subset\mathfrak{p}$. Since
$\mathfrak{h}\not\subset\mathfrak{p}$, there exists a positive root
$\alpha\in\Phi^{+}$ such that $\mathfrak{g}_{-\alpha}\cap\mathfrak{h}\neq 0.$
Let $E_{-\alpha}\in\mathfrak{g}_{-\alpha}\cap\mathfrak{h}$ be a non-zero
element. For any non-zero $E_{\alpha}\in\mathbb{R}\Theta(E_{-\alpha})$ the
subalgebra
$\mathfrak{s}:=\mathbb{R}E_{\alpha}+\mathbb{R}E_{-\alpha}+\mathbb{R}[E_{\alpha},E_{-\alpha}]$
is isomorphic to $\mathfrak{sl}_{2}(\mathbb{R})$ [Kn02, p. 68]. Note that
since $\mathfrak{n}\subset\mathfrak{h}$, we automatically have
$E_{\alpha}\in\mathfrak{h}$, so $\mathfrak{s}$ is a subalgebra of
$\mathfrak{h}$. Let $Y_{\alpha}=[E_{\alpha},E_{-\alpha}].$ The rank of $G$ is
$1$ so $\mathfrak{a}=\mathbb{R}Y_{\alpha}$.

We recall that the representations
$\mathfrak{s}\simeq\mathfrak{sl}_{2}(\mathbb{R})$ have symmetric weight space
decomposition with respect to $\mathfrak{a}$. For every
$\xi\in\mathfrak{a}^{*}$ we have
$\dim(\mathfrak{h}\cap\mathfrak{g}_{\xi})=\dim(\mathfrak{h}\cap\mathfrak{g}_{-\xi})$.
Therefore

$\dim\mathfrak{n}=\dim(\mathfrak{h}\cap\mathfrak{n})=\sum_{\beta\in\Sigma^{+}}\dim(\mathfrak{h}\cap\mathfrak{g}_{\beta})=\dim(\mathfrak{h}\cap\mathfrak{n}^{-})=\dim\mathfrak{n}^{-},$

so $\mathfrak{n}^{-}\subset\mathfrak{h}$. This proves that
$\mathfrak{n}+\mathfrak{n}^{-}\subset\mathfrak{h}$.

Claim. $\mathfrak{n}+\mathfrak{n}^{-}$ generates $\mathfrak{g}$. To prove the
claim we pass to the complexification $\mathfrak{g}_{\mathbb{C}}$. It will be
enough to show that $\mathfrak{n}_{\mathbb{C}}+\mathfrak{n}^{-}_{\mathbb{C}}$
generate $\mathfrak{g}_{\mathbb{C}}$. Choose a maximal Cartan subalgebra
$\mathfrak{b}$ of $\mathfrak{m}$ and let
$\mathfrak{c}=\mathfrak{a}+i\mathfrak{b}$. Then, $\mathfrak{c}_{\mathbb{C}}$
is a Cartan subalgebra of $\mathfrak{g}_{\mathbb{C}}$ and all the roots of
$\mathfrak{c}$ in $\mathfrak{g}_{\mathbb{C}}$ are real. Let $\Phi$ be the root
system of $\mathfrak{c}$ and finally let $V_{0}\subset\mathfrak{c}^{*}$ be the
subspace

$V_{0}:=\\{\xi\in\mathfrak{c}^{*}|\xi(Y_{\alpha})=0\\}.$

The centralizer of $\mathfrak{a}$ is
$\mathfrak{a}_{\mathbb{C}}+\mathfrak{m}_{\mathbb{C}}$ so the roots of
$\mathfrak{c}$ in $\mathfrak{m}_{\mathbb{C}}$ are precisely those that vanish
on $\mathfrak{a}$. Therefore

$\mathfrak{n}_{\mathbb{C}}+\mathfrak{n}^{-}_{\mathbb{C}}=\sum_{\xi\in\Phi\setminus
V_{0}}\mathfrak{g}_{\mathbb{C},\xi}.$

By [Kn02, Prop. 4.1.(g)], we have $\dim\mathfrak{g}_{\mathbb{C},\lambda}=1$
for every root $\lambda\in\Phi$. Moreover, we have [Kn02, p.61]222These
identities are true only for root systems of complex semisimple algebras,
which is why we had to pass to the complexification of $\mathfrak{g}$.

(3)
$[\mathfrak{g}_{\mathbb{C},\beta},\mathfrak{g}_{\mathbb{C},\gamma}]=\begin{cases}\mathfrak{g}_{\mathbb{C},\beta+\gamma}&\textrm{if
}\beta+\gamma\in\Phi,\\\ \mathbb{C}X_{\beta}&\textrm{if }\gamma=-\beta,\\\
0&\textrm{otherwise,}\end{cases}$

where $X_{\beta}$ is an element of $\mathfrak{c}$ such that
$\xi(X_{\beta})=\langle\xi,\beta\rangle$ for every $\xi\in\mathfrak{c}^{*}$.
By Lemma 3.7, $\Phi\setminus V_{0}$ generates $\Phi$. By (3) the set of roots
in the Lie algebra generated by
$\mathfrak{n}_{\mathbb{C}}+\mathfrak{n}_{\mathbb{C}}$ is closed under taking
reflections so it must be the whole root system. Using (3) once again we
deduce that $\mathfrak{n}_{\mathbb{C}}+\mathfrak{n}^{-}_{\mathbb{C}}$
generates $\mathfrak{g}_{\mathbb{C}}$. The claim is proved.

It follows that $\mathfrak{h}=\mathfrak{g}$, so $H=G$. ∎

###### Proof of Theorem 5.1.

Assume the stabilizers are not discrete almost surely. By ergodicity, the
dimension of the Lie algebra of $\overline{}{\rm Stab_{G}}(x)$ is almost
surely equal to some constant $k$. The action is non-trivial so $k<\dim G$.
Let $\mathcal{G}_{k,\mathfrak{g}}$ be the Grassmannian of $k$-planes in
$\mathfrak{g}$ and let

$\pi\colon\mathcal{G}_{k,\mathfrak{g}}\ni
W\to\left[\bigwedge\nolimits^{k}W\right]\in\mathbb{P}^{1}\left(\bigwedge\nolimits^{k}\mathfrak{g}\right)$

be the Plücker embedding. The image of the Grassmannian is a closed subset of
$\mathbb{P}^{1}(\bigwedge^{k}\mathfrak{g})$. The map $x\mapsto\pi({\rm
lie}(\overline{{\rm Stab_{G}x}}))\in\mathbb{P}^{1}(\bigwedge^{k}\mathfrak{g})$
is $G$-equivariant. Let $\mu^{\prime}$ be the pushforward of $\mu$ to
$\mathbb{P}^{1}(V)$. By Lemma 5.2, $(\mathbb{P}^{1}(V),\mu^{\prime})$ is
supported on a single orbit $G[v]$, with $H:={\rm Stab_{G}}([v])\supset AN$.
The group $G$ is simple, so no $k$-dimensional subspace of $\mathfrak{g}$ can
be fixed by $G$. Since the orbit $G[v]$ is contained in the image of the
Plücker embedding we deduce that $H$ is a proper subgroup of $G$. By Lemma 5.3
$H$ is a subgroup of $P$, so $(\mathbb{P}^{1}(V),\mu^{\prime})$ admits
$(G/P,\mu_{P})$ as a factor.

Assume now that ${\rm Stab}_{G}(x)$ is not Zariski dense almost surely. The
stabilizers must be infinite because by Lemma 3.10 there are no
$\nu$-stationary measures on the conjugacy classes of finite subsets of
$G\setminus\\{1\\}$. Therefore the stabilizers must have Zariski closures of
dimension between $1$ and $\dim\mathfrak{g}-1$. By ergodicity, there exists
$1\leq k<\dim\mathfrak{g}$ such that the Zariski closure $\overline{{\rm
Stab}_{G}x}^{Z}$ is $k$-dimensional almost surely. Consider the map
$x\mapsto\pi({\rm lie}(\overline{{\rm
Stab_{G}x}}^{Z}))\in\mathbb{P}^{1}(\bigwedge^{k}\mathfrak{g})$. Arguing as in
the paragraph above we prove that $(X,\mu)$ admits $(G/P,\mu_{P})$ as a
factor. ∎

##  6\. Stiffness of discrete stationary random subgroups for higher rank
groups

In this section we establish a stiffness result for stationary measures on the
space of discrete subgroups. In particular we show that every such measure
which is ‘irreducible’ with respect to the rank one factors of $G$ is
invariant (see Theorem 1.7). This result is a consequence of a decomposition
theorem (Theorem 6.5) which extends the results of §4 from invariant to
stationary measures. The key ingredient in the proof of Theorem 6.5 is the
following result, due to Nevo and Zimmer [NZ02].

######  Theorem 6.1 (Nevo-Zimmer [NZ02]).

Let $G$ be a higher rank semisimple Lie group. Let $\nu$ be a smooth
probability measure on $G$ and let $(X,\mu)$ be a probability $\nu$-stationary
action of $G$. Then either

  * •

$\mu$ is $G$-invariant,

  * •

there exists a proper parabolic subgroup $Q\subset G$ and a measure preserving
$G$-equivariant map $\pi\colon(X,\mu)\to(G/Q,\nu_{Q}),$ where $\nu_{Q}$ is the
unique $\nu$-stationary measure on $G/Q$, or

  * •

$(X,\mu)$ has an equivariant factor space, on which $G$ acts via a rank one
factor group.

In what follows, $P$ will be a minimal parabolic subgroup of $G$.

######  Lemma 6.2.

Let $Q$ be a parabolic subgroup of $G$. Let $Q=LN$ be a Levi decomposition of
$Q$ and let $A_{L}$ be the centre of $L$. Then, any discrete $A_{L}$-invariant
random subgroup of $Q$ is contained in $L$ almost surely.

###### Proof.

Let $a\in A_{L}$ be an element such that the norm of the restriction of
$\text{Ad}(a^{-1})$ to the Lie algebra of $N$ is less than $1$. It follows
that for any two identity neighbourhoods $U_{1},U_{2}\subset N$ in $N$ such
that $U_{2}$ is bounded there is $n_{0}\in{\mathbb{N}}$ such that
$\text{Ad}(a^{-n_{0}})(U_{2})\subset U_{1}$.

Let $\lambda$ be a discrete $A_{L}$-invariant random subgroup of $Q$. Suppose
by way of contradiction that $\lambda(\\{\Gamma\subset L\\})<1$. Then there is
a bounded identity neighbourhood $V_{2}$ in $Q$ such that $\lambda(\Omega)>0$
where $\Omega:=\\{\Gamma:\Gamma\cap V_{2}\setminus L\neq\emptyset\\}$.
Furthermore, we may take $V_{2}$ to be of the form $V_{2}=W\cdot U_{2}$ where
$W\subset L$ and $U_{2}\subset N$ and suppose that they are preserved by
$a^{-1}$ (e.g. we can suppose that $\log U_{2}$ is a norm ball in the Lie
algebra of $N$). Since $\lambda$ is discrete we can chose a small identity
neighborhood $U_{1}\subset N$ so that setting $V_{1}=W\cdot U_{1}$ we have

$\epsilon:=\lambda(\\{\Gamma\in\Omega:\Gamma\cap(V_{1}\setminus
L)=\emptyset\\})>0.$

Choosing $n_{0}$ as above we get that
$\lambda(\Omega^{a^{n_{0}}})\leq\lambda(\Omega)-\epsilon$ in contrast with the
assumption that $\lambda$ is $A_{L}$-invariant. ∎

######  Lemma 6.3.

Let $G$ be a centre-free complex semisimple Lie group, let $Q$ be a proper
parabolic subgroup. The intersection of all Levi subgroups of $Q$ is the
product of the simple factors of $G$ contained in $Q$.

###### Proof.

We can quotient $G$ by the product of all simple factors contained in $Q$. In
this way we can assume that $Q$ contains no simple factors of $G$. Every
parabolic subgroup in $G$ is a product of parabolic subgroups in the simple
factors, so the lemma reduces to the case where $G$ is simple.

Assume from now on that $G$ is simple. Let $J$ be the intersection of all Levi
subgroups of $Q$. We need to show that $J=\\{1\\}$. Let $L$ be a Levi subgroup
of $Q$ and let $N$ be the unipotent radical of $Q$. Let $P\subset Q$ be a
Borel subgroup with a Levi subgroup $A\subset L$. Then $A$ is a maximal torus
of $G$. Write $\mathfrak{g,q,l,n,p,a}$ for the corresponding Lie algebras. Let
$\Phi$ be the root system of $\mathfrak{g}$ with respect to $\mathfrak{a}$.
Let $\Phi^{+}$ be the set of positive roots corresponding to $P$ and let
$\Pi\subset\Phi^{+}$ be the subset of simple roots. Let $W$ be the Weyl group.
Finally let $F\subset\Pi$ be the subset of simple roots lying in
$\mathfrak{q}$. Write $E\subset\mathfrak{a}^{*}$ for the space spanned by $F$
and let $S=\Phi\cap E$. Then

$\mathfrak{l}=\sum_{\lambda\in
S}\mathfrak{g}_{\lambda},\qquad\mathfrak{n}=\sum_{\begin{subarray}{c}\lambda\in\Phi^{+}\setminus
S\end{subarray}}\mathfrak{g}_{\lambda},\qquad\mathfrak{q}=\mathfrak{l}+\mathfrak{n}=\sum_{\lambda\in
S\cup\Phi^{+}}\mathfrak{g}_{\lambda}.$

Let $R:=\Phi^{+}\setminus S$. By Lemma 3.7, $R$ spans $\mathfrak{a}^{*}$. Let
$A^{\prime}\subset A$ be a maximal torus of $J$. Since both $J$ and $N$ are
normal in $Q$ we have $[J,N]\subset J\cap N\subset L\cap N=\\{1\\}$. The torus
$A^{\prime}$ commutes with $N$ so the roots in $R$ vanish on its Lie algebra
$\mathfrak{a}^{\prime}$. As $R$ spans $\mathfrak{a}^{*}$ we get
$\mathfrak{a}^{\prime}=0$. As a normal subgroup of $L$, $J$ must be reductive,
so triviality of maximal tori implies that $J$ is finite. Since $J$ is finite
and normal in $P$ it must be in the centre, so $J=\\{1\\}$. ∎

######  Lemma 6.4.

Let $G$ be a centre-free semisimple Lie group with proper parabolic subgroup
$P\subset Q$. Let $\Lambda\subset Q$ be a discrete $P$-invariant random
subgroup of $Q$. Then there is a proper semisimple factor $H$ of $G$ such that
$\Lambda$ is supported on discrete subgroups of $H$ almost surely.

###### Proof.

Let $L\leq Q$ be a Levi subgroup of $Q$, such that the centre $A$ of $L$ is
contained in $P$. By Lemma 6.2, $\Lambda\subset L$ almost surely. Since
$\Lambda$ is $P$-invariant we can upgrade this inclusion to
$\Lambda\subset\bigcap_{p\in P}L^{p}=\bigcap_{q\in Q}L^{q}$. Let $J$ be the
intersection of all Levi subgroups of $Q$. The set of real points of $Q$ is
Zariski dense so $J$ coincides with the real points of the intersection of all
complex Levi subgroups of $Q(\mathbb{C})$. By Lemma 6.3 we deduce that
$\Lambda$ is supported on $\text{Sub}_{d}(H)$ where $H=\bigcap_{q\in Q}L^{q}$
is a proper semisimple factor of $G$ almost surely. ∎

The following decomposition theorem generalizes Theorem 4.1 from invariant
random subgroups to stationary discrete subgroups:

######  Theorem 6.5.

Let $G$ be a connected centre-free semisimple Lie group without compact
factors and $\mu$ a discrete $\nu$-stationary random subgroup of $G$. Then $G$
decomposes to a product of three semisimple factors $G=G_{\mathcal{I}}\times
G_{\mathcal{H}}\times G_{\mathcal{T}}$ such that

  1. (1)

$\mu$ projects to an IRS in $G_{\mathcal{I}}$ for which all the irreducible
factors are of rank at least $2$.

  2. (2)

$G_{\mathcal{H}}$ is a product of rank one factors and $\mu$ projects
discretely to every factor of $G_{\mathcal{H}}$.

  3. (3)

$\mu$ projects trivially to $G_{\mathcal{T}}$.

Furthermore, the intersection of a random subgroup with every simple factor of
$G_{\mathcal{H}}$ as well as with every irreducible factor of
$G_{\mathcal{I}}$ is almost surely Zariski dense in that factor.

By the irreducible factors of $G_{\mathcal{I}}$ we mean the irreducible
factors associated with the decomposition of IRS as in Theorem 4.1. The
subscripts $\mathcal{I},\mathcal{H},\mathcal{T}$ stands for invariant,
hyperbolic and trivial (respectively).

###### Proof of Theorem 6.5.

We will argue by induction on the number of simple factors of $G$. If $G$ is
simple then we may suppose $\text{rank}(G)\geq 2$ since for rank one groups
the statement trivially holds. If $\mu$ is invariant then it is either trivial
or Zariski dense by the Borel density theorem for IRS.

Suppose by way of contradiction that $G$ is simple of higher rank and $\mu$ is
not invariant. Let $\pi\colon(\text{Sub}(G),\mu)\to(G/Q,\mu_{Q})$ be the
measure preserving $G$-equivariant map afforded by Theorem 6.1. Let $P\subset
Q$ be a minimal parabolic.

Let $\mathcal{P}(\text{Sub}(G))$ be the space of Borel probability measures on
$\text{Sub}(G)$. As a first step we will construct a $G$-equivariant map

$\kappa\colon(G/P,\mu_{P})\to\mathcal{P}(\text{Sub}(G)),$

such that $\mu=\int_{G/P}\kappa(gP)d\mu_{P}(gP).$ Consider the map $\psi\colon
G\to\mathcal{P}(\text{Sub}(G))$ given by $\psi(g):=g_{*}\mu$. Then, $\psi$ is
a $G$-equivariant, bounded $\nu$-harmonic function on $G$. The pair
$(G/P,\mu_{P})$ is the Furstenberg–Poisson boundary, so there exists a unique
measurable function $\kappa\colon(G/P,\mu_{P})\to\mathcal{P}(\text{Sub}(G))$
such that $\mu=\int_{G/P}\kappa(gP)d\mu_{P}(gP).$ The uniqueness implies that
the map is $G$-equivariant. This also implies that $\kappa(gP)$ is $gPg^{-1}$
invariant $\mu_{P}$-almost surely.

Consider the composition
$\pi_{*}\circ\kappa\colon(G/P,\mu_{p})\to\mathcal{P}(G/Q).$ This map is
$G$-equivariant so the measure $\pi_{*}\circ\kappa(gP)$ is
$gPg^{-1}$-invariant almost surely. The unique probability measure on $G/Q$
with this property is $\delta_{gQ}$. Hence $\pi\circ\kappa(gP)=gQ$ almost
surely.

By comparing the stabilizers we deduce that for $\mu_{P}$ almost every $gP\in
G/P$, the measure $\kappa(gP)$ is supported on the set of discrete subgroups
of $gQg^{-1}$. The measure $\kappa(gP)$ is $gPg^{-1}$-invariant, so it is the
distribution of a $gPg^{-1}$-invariant random discrete subgroup of $gQg^{-1}$.
By Lemma 6.4 we get $\kappa(gP)=\delta_{\\{1\\}}$ almost surely. Since
$\mu=\int_{G/P}\kappa(gP)d\mu_{P}(gP)$ we deduce that $\mu=\delta_{\\{1\\}}$,
a contradiction. This completes the proof of the theorem when $G$ is simple.

Suppose now that $G=\prod_{i=1}^{n}G_{i}$ has $n>1$ factors. In view of Remark
3.1 we may suppose that $\nu=\nu_{1}*\cdots*\nu_{n}$ where $\nu_{i}$ is a
probability measure on $G_{i}$. If $\mu$ is invariant the theorem follows from
Theorem 4.1 and Corollary 4.4. If $\mu$ admits a parabolic factor
$(G/Q,\mu_{Q})$ then arguing as in the simple case above we deduce from Lemma
6.4 that $\mu$ is supported on discrete subgroups of $H$ for some proper
semisimple factor $H\lhd G$. Then we deduce the result from the induction
hypothesis.

Thus we are left with the case where $\mu$ is not invariant and does not admit
a $G/Q$-factor. Then it follows from Theorem 6.1 and Theorem 5.1 that for some
rank one factor, say, $G_{1}$, the projection of a $\mu$-random subgroup is
almost surely discrete and Zariski dense in $G_{1}$. Consider the intersection
with $H=G_{2}\times\ldots\times G_{n}$. Since the intersection is a
$\nu_{2}*\cdots*\nu_{n}$ stationary random subgroup it follows from the
induction hypothesis that there is a decomposition $H=H_{\mathcal{I}}\times
H_{\mathcal{H}}\times H_{\mathcal{T}}$ as in the statement of the theorem. Let
$\Delta$ denote the $\mu$-random subgroup. Since the intersection
$\Delta\cap(H_{\mathcal{I}}\times H_{\mathcal{H}})$ is almost surely Zariski
dense in $H_{\mathcal{I}}\times H_{\mathcal{H}}$ and normalized by the
projection of $\Delta$ to $H_{\mathcal{I}}\times H_{\mathcal{H}}$, it follows
that this projection is discrete as well. Thus the projection of a
$\mu$-random subgroup to $G_{1}\times H_{\mathcal{I}}\times H_{\mathcal{H}}$
is discrete almost surely. Applying Lemma 3.13 we deduce that a random
subgroup projects trivially to $H_{\mathcal{T}}$. Furthermore, by the same
reasoning we see that the projection of $\Delta$ to every irreducible factor
of $H_{\mathcal{I}}$ as well as to every simple factor of $H_{\mathcal{H}}$ is
discrete. Finally, consider the intersection of the random subgroup with
$G_{1}$. By Lemma 3.13 this intersection is non-trivial and since it is
normalized by the projection to $G_{1}$ which by assumption is Zariski dense
we deduce that the intersection with $G_{1}$ is also Zariski dense. Thus our
desired decomposition is given by

$G_{\mathcal{I}}=H_{\mathcal{I}},~{}G_{\mathcal{H}}=G_{1}\times
H_{\mathcal{H}},~{}G_{\mathcal{T}}=H_{\mathcal{T}}.$

∎

We are now in a position to deduce:

######  Theorem 6.6 (Theorem 1.7 of the introduction).

Let $G$ be a connected centre-free semisimple Lie group without compact
factors and real rank at least two. Let $\mu$ be a $\nu$-stationary measure on
$\text{Sub}_{d}(G)$. Suppose that $\mu$-almost every random subgroup
intersects trivially every rank one factor of $G$. Then $\mu$ is invariant.

###### Proof of Theorem 1.7.

The assumption on the rank one factors implies that in the decomposition of
$G$ according to $\mu$ given by Theorem 6.5 the factor $G_{\mathcal{H}}$ is
trivial. ∎

######  Remark 6.7.

The assumption that for every rank one factor the intersection is trivial can
be replaced by the assumption that the intersection is not Zariski dense in
that factor almost surely. This applies also to Theorem 7.1 below.

We remark that the higher rank assumption in Theorem 1.7 is necessary. The
following construction demonstrates how to produce a stationary non-invariant
measure on the space of discrete subgroups:333Nevo and Zimmer constructed
stationary actions of rank one groups on compact spaces which admit no
invariant probability measure [NZ99, Theorem B]. However in their
construction, the pushforward of every stationary measure to the space of
discrete subgroups via the stabilizer map collapses to an invariant measure on
$\text{Sub}_{d}(G)$.

######  Example 6.8.

Let $T$ be a $3$ regular tree and consider $X$ to be its grandfather graph
according to a chosen point at the visual boundary $\partial T$. That is, $X$
is obtained from $T$ by adding edges between any two points of distance 2
which lie on a ray to the chosen point at infinity. Then $X$ is an $8$ regular
graph, obtained from $T$ by adding two disjoint $5$-regular trees and is hence
$8$ colorable using $3$ colors for $T$ and another $5$ for the new edges. By
doubling every edge and giving it the two possible orientations we obtain a
Schreier graph $F_{8}/H$ associated with some ‘uniformly slim’ subgroup $H\leq
F_{8}$. Note that the closure of the conjugacy class $\mathcal{C}$ of $H$ in
$F_{8}$ consists of groups whose Schreier graph is a similar such oriented
colouring of $X$. Since the graph $X$ ‘remembers’ the chosen point at the
boundary of $T$, we deduce that there is no $F_{8}$-invariant probability
measure on $\mathcal{C}$. Indeed $F_{8}$ acts transitively on $T$ and hence
preserves no probability measure on $\partial(T)$. Embed $F_{8}$ as a lattice
in $\text{SL}(2,{\mathbb{R}})$ and let $\Lambda$ be the image of $H$. Then
every weak-$*$ limit of the random walk
$\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(i)}*\delta_{\Lambda}$ is a stationary
measure on $\text{Sub}_{d}(\text{SL}(2,{\mathbb{R}}))$ which in view of [NZ99,
Lemma 6.1] cannot be invariant. Similar constructions can be made for every
rank one group that admits a lattice which projects on $F_{2}$.

##  7\. Stück–Zimmer theorem for stationary measures

The decomposition result, Theorem 6.5, allows to extend the celebrated
Stück–Zimmer theorem to stationary measure with discrete stabilizers.

######  Theorem 7.1 (Theorem 1.9 of the introduction).

Let $G$ be a connected centre-free semisimple Lie group without compact
factors. Suppose that $G$ has real rank at least $2$ and Kazhdan’s property
(T). Let $\mu$ be an ergodic $\nu$-stationary measure on $\text{Sub}_{d}(G)$.
Suppose that $\mu$-almost every random subgroup intersects trivially every
rank one factor of $G$. Then there is a semisimple factor $H\lhd G$ and a
lattice $\Gamma\leq H$ such that $\mu=\mu_{\Gamma}$.

We will rely on the following proper ergodicity result:

######  Lemma 7.2.

Let $G=G_{1}\times G_{2}$ where $G_{1},G_{2}$ are locally compact second
countable groups. Let $X$ be a compact, second countable $G$-space with
$G_{1}$-invariant probability measure $\mu$ such that ${\rm Stab}_{G}(x)$ is
discrete and has a dense projection onto $G_{2}$ for $\mu$-almost every $x\in
X$. Then $\mu$ is $G$-invariant.

###### Proof.

In view of the ergodic decomposition of probability measure preserving
actions, the lemma clearly reduces to the ergodic case, so let us assume that
$\mu$ is $G_{1}$-ergodic. Choose a smooth symmetric probability measure $\eta$
on $G_{1}$ whose support generates $G_{1}$. Write $\eta^{g_{1}}$ for the
measure $\eta^{g_{1}}(A)=\eta(g_{1}^{-1}Ag_{1})$. Since the support of $\eta$
generates $G_{1}$, the measure $\mu$ is ergodic with respect to the random
walk on $X$ induced by $\eta^{g_{1}}$, for every $g_{1}\in G_{1}$. We say that
a point $x\in X$ is $\eta^{g_{1}}$-generic for $\mu$ if

(4) $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n-1}\int
f(gx)d(\eta^{g_{1}})^{\ast
i}(g)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n-1}\int
f(g_{1}^{-1}gg_{1}x)d\eta^{\ast i}(g)=\int fd\mu(x),$

for all continuous functions $f$. By the pointwise ergodic theorem for random
walks we know that the set of $\eta^{g_{1}}$-generic points has full measure
with respect to $\mu$. It follows that the set

$\\{(g_{1},x)|x\textrm{ is not }g_{1}\eta g_{1}^{-1}\textrm{-generic}\\}$

has zero measure. By Fubini’s theorem we deduce that there is a subset
$X^{\prime}\subset X$ with $\mu(X^{\prime})=1$, such that every $x\in
X^{\prime}$ satisfies the following property. For almost every $g_{1}\in
G_{1}$

(5) $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n-1}(\eta^{g_{1}})^{\ast
i}\delta_{x}=\mu.$

in the weak-* topology.

Take a point $x\in X^{\prime}$ such that $\Gamma:={\rm Stab}_{G}(x)$ is
discrete and has a dense projection onto $G_{2}$. Since $\Gamma$ is countable,
we may fix $g_{1}\in G$ such that (5) holds for $g_{1}\gamma_{1}$, for every
$\gamma=(\gamma_{1},\gamma_{2})\in\Gamma$. Note that $\eta^{\gamma
g}=\eta^{\gamma_{1}g_{1}}$. Comparing (5) for $g_{1}$ and $\gamma_{1}g_{1}$
and using the fact that $\gamma^{-1}x=x$ we find that

$\displaystyle\mu=$
$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n-1}(\eta^{\gamma_{1}g_{1}})^{\ast
i}\delta_{x}=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n-1}(\eta^{\gamma
g_{1}})^{\ast i}\delta_{x}$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n-1}\gamma_{*}(\eta^{g_{1}})^{\ast
i}\delta_{\gamma
x}=\gamma_{*}\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n-1}(\eta^{g_{1}})^{\ast
i}\delta_{x}=\gamma_{*}\mu.$

This means that $\Gamma G_{1}\subset{\rm Stab}_{G}\mu$. The action of $G$ on
the set of probability measures on $X$ is continuous, so ${\rm
Stab}_{G}\mu\supset\overline{\Gamma G_{1}}=G.$ ∎

######  Corollary 7.3.

Let $G_{1}$ and $G_{2}$ be locally compact second countable groups. Let $\mu$
be an ergodic discrete invariant random subgroup in $G_{1}\times G_{2}$ with
dense projections to $G_{2}$. Then $\mu$ is $G_{1}$-ergodic.

###### Proof.

Let $\mu$ be an ergodic discrete invariant random subgroup in $G_{1}\times
G_{2}$ with dense projections to $G_{2}$. In view of [7s17, Theorem 2.6] $\mu$
can be realised as the stabilizer of some compact, second countable $G$-space
$X$. Abusing notations we will denote the measure on $X$ by $\mu$ as well.
Consider the ergodic decomposition of $(X,\mu)$ with respect to the
$G_{1}$-action. By Lemma 7.2 the $G_{1}$-ergodic components of $\mu$ are
$G_{2}$-invariant. Since $\mu$ is $G_{1}\times G_{2}$-ergodic it follows that
the decomposition is trivial. This means that $\mu$ is $G_{1}$-ergodic. ∎

Thus we have established:

######  Corollary 7.4.

Under the assumptions of Corollary 4.4, the IRS $\overline{\mu}_{j}$ are
properly ergodic, that is, $\overline{\mu}_{j}$ is ergodic with respect to
every simple factor of $H_{j}$.

Recall the main result of [SZ94] (see also [7s17, §4]):

######  Theorem 7.5 (Stück–Zimmer [SZ94]).

Let $G$ be a centre-free semisimple Lie group of real rank at least $2$ and
with Kazhdan’s property $(T)$. Suppose that $G$, as well as every rank one
factor of $G$, acts ergodically and faithfully preserving a probability
measure on a space $X$. Then there is a normal subgroup $N\lhd G$ and a
lattice $\Gamma<N$ such that for almost every $x\in X$ the stabilizer of $x$
is conjugate to $\Gamma$.

The following is a straightforward conclusion of the combination of Theorem
4.1, Corollary 4.4, Corollary 7.4, and the Stück Zimmer theorem:

######  Proposition 7.6.

Let $G=G_{1}\times\cdots\times G_{n}$ be a connected centre-free semisimple
Lie group without compact factors. Suppose that $G$ has Kazhdan’s property
(T). Let $\mu$ be an ergodic Zariski dense discrete invariant subgroup in $G$,
and suppose that the projection of a $\mu$-random subgroup to every rank one
simple factor of $G$ is non-discrete. Then $\mu$ is supported on lattices in
$G$.

######  Remark 7.7.

Note that in the context of Proposition 7.6 the three invariant random
subgroups $\mu,\tilde{\mu}$ and $\overline{\mu}$ introduced after Theorem 4.1
are commensurable in the obvious sense.

######  Remark 7.8.

In view of [HT16, Corollary 1.4] it is enough to suppose in Proposition 7.6
that $\mu$ is irreducible in the sense of Theorem 4.1 and one of the factors
of $G$ has property (T).

We can now complete the proof of Theorem 1.9. Suppose that $\mu$ is not
$\delta_{\\{1\\}}$. As in the proof of Theorem 1.7 we get that
$G=G_{\mathcal{I}}\times G_{\mathcal{T}}$. Since all the irreducible factors
of $G_{\mathcal{I}}$ are of rank at least $2$, the result follows from
Proposition 7.6. ∎

##  8\. Margulis conjecture for simple Lie groups of rank at least $2$.

When applied to simple Lie groups of real rank at least $2$, Theorem 1.9 reads
as:

######  Theorem 8.1.

Let $G$ be a centre-free higher rank simple Lie group. Let $\nu$ be a smooth
probability measure on $G$. Let $\mu$ be an ergodic $\nu$-stationary random
discrete subgroup of $G$. Then $\mu=\delta_{\\{1\\}}$, or there exists a
lattice $\Gamma\subset G$ such that $\mu=\mu_{\Gamma}$.

This allows us to prove:

######  Theorem 8.2.

Let $G$ be a centre-free higher rank simple Lie group. Let $\Lambda\leq G$ is
a discrete subgroup of infinite covolume. Then

$\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(i)}*\delta_{\Lambda}\to\delta_{\\{1\\}}.$

###### Proof.

Let $\mu$ be a stationary limit of
$\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(i)}*\delta_{\Lambda}$. Then $\mu$ is
supported on the closure of the conjugacy class of $\Lambda$ in
$\text{Sub}(G)$. Moreover, in view of Theorem 2.2, $\mu(\text{Sub}_{d}(G))=1$.
Since $G$ is simple of higher rank, lattices in $G$ are locally rigid. Thus
the closure of the conjugacy class of $\Lambda$ in $\text{Sub}(G)$ contains no
lattices. It follows from Theorem 8.1 that $\mu=\delta_{\\{1\\}}$. ∎

Theorem 8.2 can be reformulated as follows: Let $M=\Lambda\backslash G/K$ be a
locally symmetric space of infinite volume, where $G$ is a simple Lie groups
of rank at least $2$. Consider the $\nu_{G}$-random walk on $M$ starting at
some point $x_{0}$. Then for every $r>0$, the random walk will almost
certainly eventually spend most of the time in the $r$-thick part. More
precisely, let $\tilde{x}_{0}\in G/K$ be a lift of $x_{0}$ and let
$m_{n}\in\text{Prob}(M)$ be the pushforward of
$\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(i)}*\delta_{\tilde{x}_{0}}$ via the
covering map. Then we have:

######  Corollary 8.3.

For every $r>0$ and $\epsilon>0$ there is $N$ such that, for $n\geq N$,
$m_{n}(M_{\geq r})>1-\epsilon$. ∎

As a straightforward consequence of Theorem 8.2, we obtain the following
equivalence:

######  Theorem 8.4.

Let $G$ be a centre-free higher rank simple Lie group. Let $\Delta\leq G$ be a
discrete subgroup. Then $\Delta$ is uniformly slim if and only if $\Delta$ is
a lattice in $G$.

###### Proof.

Latices are obviously uniformly slim. On the other hand if $\Delta\leq G$ is
of infinite covolume, it follows from Theorem 8.2 that $\\{1\\}$ is a
conjugate limit of $\Delta$. Thus $\Delta$ is not uniformly slim. ∎

##  9\. Uniformly slim subgroups of semisimple groups

Throughout this section we will assume that $G=G_{1}\times\cdots\times G_{m}$
is a connected centre-free semisimple Lie group without compact factors. The
$G_{i}$’s are the simple factors of $G$ and we suppose that $m\geq 2$. Since
the normal subgroup theorem requires higher rank and the Stück–Zimmer theorem
requires in addition property (T), the rank one factors play a special role.

Recall that there are three infinite families and one exceptional simple real
Lie group of rank one. Among them $\text{Sp}(n,1),~{}n\geq 2$ and $F_{4(-20)}$
have Kazhdan’s property (T) while the real and complex hyperbolic groups
$\text{PO}(n,1)$ and $\text{PU}(n,1),~{}n\geq 2$ do not have property (T).

######  Definition 9.1.

We shall say that a semisimple group $G$ is of type (L) if every simple factor
of $G$ is either $\text{PO}(n,1)$ or $\text{PU}(n,1),~{}n\geq 2$.

######  Theorem 9.2.

Let $G$ be a connected centre-free semisimple Lie group without compact
factors. Let $\Lambda$ be a discrete subgroup of $G$ and suppose that for
every nontrivial normal subgroup $H\lhd G$, the intersection $\Lambda\cap H$
is not a lattice in $H$. Let $\mu$ be an ergodic component of a stationary
limit of $\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(i)}*\delta_{\Lambda}$. Then
there are (possibly trivial) normal subgroups $H_{1},H_{2}\lhd G$, where
$H_{1}$ is a product of rank one factors, $H_{2}$ is of type (L) and
$H_{1}\cap H_{2}=\\{1\\}$, such that $\mu$-almost surely:

  1. (1)

the random group is contained in $H_{1}\times H_{2}$ and projects discretely
to $H_{1}$ and to $H_{2}$,

  2. (2)

the projection to $H_{2}$ is an IRS whose irreducible factors are higher rank
and the intersection with each factor is thin (i.e. Zariski dense of infinite
co-volume in the factor),

  3. (3)

the intersection with every simple factor of $H_{1}$ is Zariski dense, and

  4. (4)

the intersection with every simple factor of $H_{1}$ with property (T) is
thin.

######  Remark 9.3.

It is a famous open problem whether the analog of the Stück–Zimmer theorem
holds for higher rank groups of type (L). A positive solution to this problem
would imply that the group $H_{2}$ in Theorem 9.2 is trivial. On the other
hand the group $H_{1}$ may not be trivial even if $\Lambda$ is not uniformly
slim. Using ideas from [7s20] one can construct a non-uniformly slim subgroup
of a rank one group whose stationary limit is supported on uniformly slim
subgroups.

######  Remark 9.4.

For certain specific cases we can say more about the stationary limit. For
instance if $\Lambda$ does not contain large commuting subgroups then so does
a random group with respect to a stationary limit. Such data eliminates some
of the possibilities that the general case allows. In the special case that
$\Lambda$ is an infinite index subgroup of an irreducible higher rank lattice
and $G$ has property (T), it is easy to deduce that the stationary limit must
be $\delta_{\\{1\\}}$.

######  Lemma 9.5.

Let $H\lhd G$ be a normal subgroup of $G$ and let $\Gamma\leq H$ be a lattice.
Suppose that either $\text{rank}(H)\geq 2$ and $\Gamma$ is irreducible or that
$H$ has Kazhdan’s property (T). Then $\Gamma$ is locally rigid in $G$.

###### Proof.

The case where $\text{rank}(H)\geq 2$ follows from [M90, page 332] and [Rag89,
Theorem 6.7]. Therefore we may suppose that $H$ has property (T). Let
$f:\Gamma\to G$ be a deformation and consider the unitary representation of
$\Gamma$ on $L^{2}(G/H)$ where $\Gamma$ acts via $f$. If $f$ is a sufficiently
small deformation this representation has a $(\Sigma,\epsilon)$-invariant unit
vector where $\Sigma$ is a finite generating set for $\Gamma$ and
$\epsilon=\epsilon(\Sigma)>0$ is a Kazhdan constant for $\Sigma$ given by
property (T) of $\Gamma$. It follows that this representation has a nontrivial
invariant vector which implies that the projection of $f(\Gamma)$ lies in a
compact subgroup of $G/H$. By further conjugating by some element close to the
identity we may assume that it lies in a fixed compact subgroup $K\leq G/H$.
By [Wa75, Theorem 2.6] $\Gamma$ has only finitely many conjugacy classes of
representations into $K$. Thus we deduce that eventually $f(\Gamma)\leq H$.
The result now follows from local rigidity of $\Gamma$ in $H$. ∎

######  Corollary 9.6.

Let $H\lhd G$ be as in Lemma 9.5. Let $\Lambda$ be a discrete subgroup of $G$.
Suppose that $\Lambda$ admits a discrete conjugate limit which intersects $H$
by an irreducible lattice. Then $\Lambda$ intersects $H$ by a lattice.

###### Proof.

Let $\Delta$ be a discrete conjugate limit of $\Lambda$ and suppose that
$\Gamma=\Delta\cap H$ is a lattice in $H$. Fixing a finite presentation
$\langle\Sigma:R\rangle$ for $\Gamma$ we see that if $\Lambda^{g}$ is
sufficiently close to $\Delta$ then the map sending each $\sigma\in\Sigma$ to
its nearest element in $\Lambda^{g}$ extends to a homomorphism
$f_{g}:\Gamma\to\Lambda^{g}$. Thus the corollary follows from Lemma 9.5. ∎

###### Proof of Theorem 9.2.

By Theorem 2.2, $\mu$ is a discrete stationary random subgroup. Consider the
decomposition of $G$ given by Theorem 6.5, $G=G_{\mathcal{I}}\times
G_{\mathcal{H}}\times G_{\mathcal{T}}$ and set
$H_{1}=G_{\mathcal{H}},H_{2}=G_{\mathcal{I}}$. By Theorem 6.5, $H_{1}$ is a
product of rank one factors. Suppose by way of contradiction that $H_{2}$ is
not of type (L). Then by Proposition 7.6, Remark 7.7 and Remark 7.8 a random
subgroup intersects a higher rank semisimple factor of $G_{\mathcal{I}}$ by an
irreducible lattice. This is also the case if Item (2) of the theorem is not
satisfied. Similarly, if Item (4) of the theorem is not satisfied then a
random subgroup intersects by a lattice some rank one simple factor of $G$
which has property (T). In all three cases we deduce from Corollary 9.6 that
$\Lambda$ itself intersects the given factor by a lattice. This is a
contradiction to the assumption. ∎

We deduce the following classification of uniformly slim subgroups when all
the factors of $G$ are higher rank:

######  Theorem 9.7.

Let $G=G_{1}\times\cdots\times G_{n}$ be a connected centre-free semisimple
Lie group such that $\text{rank}(G_{i})\geq 2,~{}i=1,\ldots,n$. Let $\Lambda$
be a discrete subgroup. Then $\Lambda$ is uniformly slim if and only if there
is a nontrivial semisimple factor $H\lhd G$ such that $\Lambda\cap H$ is a
lattice in $H$. If $\Lambda$ is not uniformly slim then
$\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(i)}*\delta_{\Lambda}\to\delta_{\\{1\\}}$.

######  Definition 9.8.

Let $G$ be a locally compact group. A subset $D\subset G\times G$ will be
called a density tester if it satisfies the following property: A subgroup
$\Gamma\leq G$ is dense if and only if $(\Gamma\times\Gamma)\cap
D\neq\emptyset$.

Note that any connected semisimple Lie group $G$ admits a compact density
tester. Indeed, we can take two compact sets with non-empty interior $U,V$
lying in a Zassenhaus neighbourhood of $G$ where the logarithm is well defined
such that for any $u\in U,v\in V$, we have that $\log(u)$ and $\log(v)$
generate the Lie algebra of $G$. Then it follows that for any such pair the
group $\langle u,v\rangle$ is dense in $G$ (see [BG03] or [GZ02] for details).
Thus $D=U\times V$ is a density tester.

######  Definition 9.9.

Let $G=G_{1}\times\cdots\times G_{n}$ be a connected semisimple Lie group with
simple factors $G_{i}$. Let $\Gamma\leq G$ be a discrete subgroup. We will say
that $\Gamma$ is uniformly irreducible if there are:

  * •

a compact density tester $D_{i}$ for $G/G_{i}$ for $i=1,\ldots,n$

  * •

a compact set $B\subset G$

such that for every $g\in G$ and $i$, the projection of the finite set
$\Gamma^{g}\cap B$ to $G/G_{i}$ admits two points
$\gamma_{i}^{1},\gamma_{i}^{2}$ such that $(\gamma_{i}^{1},\gamma_{i}^{2})\in
D_{i}$.

######  Remark 9.10.

Obviously a uniformly irreducible subgroup is uniformly slim. It is easy to
check that every irreducible lattice in $G$ is uniformly irreducible. It is
also not hard to check that a nontrivial normal subgroup of a uniform
irreducible lattice is uniformly irreducible.

######  Theorem 9.11.

Let $G=G_{1}\times\cdots\times G_{n},~{}n\geq 2$ be a connected centre-free
semisimple Lie group without compact factors and suppose that at least one of
the simple factors of $G$ has Kazhdan’s property (T). Let $\Lambda$ be a
discrete subgroup of $G$. Then $\Lambda$ is uniformly irreducible if and only
if $\Lambda$ is an irreducible lattice in $G$.

###### Proof.

Let $\Lambda\leq G$ be a uniformly irreducible discrete subgroup. Note that
every conjugate limit of a uniformly irreducible group is uniformly
irreducible. In view of that, it follows from Theorem 2.2, Proposition 7.6 and
Remark 7.8 that every ergodic component of a stationary limit of
$\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(i)}*\delta_{\Lambda}$, must be
$\mu_{\Gamma}$ for some irreducible lattice in $G$. In view of Corollary 9.6
this implies that $\Lambda$ is an irreducible lattice in $G$. ∎

Let $G$ be as above and let $G_{i}$ be a simple factor. We shall say that a
discrete subgroup $\Lambda$ projects uniformly densely to $G_{i}$ if there is
a compact set $B\subset G$ and a compact density tester $D$ for $G_{i}$ such
that for every $g\in G$ the projection of $\Lambda^{g}\cap B$ to $G_{i}$
contains two points $\alpha,\beta$ such that $(\alpha,\beta)\in D$. The
following is a variant of Theorem 9.11. We leave the proof as an exercise.

######  Proposition 9.12.

Let $G=G_{1}\times\cdots\times G_{n},~{}n\geq 2$ be a connected centre-free
semisimple Lie group without compact factors and with Kazhdan’s property (T).
Let $\Lambda$ be a discrete uniformly slim subgroup of $G$. Suppose in
addition that $\Lambda$ projects uniformly densely to every rank one factor of
$G$. Then there is a nontrivial semisimple factor $H\lhd G$ which contains all
the rank one factors of $G$ such that $\Lambda\cap H$ is a lattice in $H$.

We shall now prove the following alternative:

######  Theorem 9.13.

Let $G=G_{1}\times\cdots\times G_{n},~{}n\geq 2$ be a connected centre-free
semisimple Lie group without compact factors and with Kazhdan’s property (T).
Let $\Lambda\leq G$ be a discrete uniformly slim group. Then exactly one of
the following holds:

  1. (i)

There is a nontrivial proper semisimple factor $H\lhd G$ such that
$\Lambda\cap H$ is a lattice in $H$.

  2. (ii)

There is a proper nontrivial semisimple factor $H\lhd G$ which is a product of
rank one simple factors, $H=\prod_{i=j}^{k}G_{i_{j}}$, and a conjugate limit
$\Delta$ of $\Lambda$, such that $\Delta$ is a discrete uniformly slim
subgroup of $H$. Furthermore $\Delta$ intersects every simple factor of $H$ in
a uniformly slim thin subgroup.

######  Lemma 9.14.

A uniformly slim discrete subgroup of a simple Lie group is Zariski dense.

###### Proof.

Let $G$ be a simple group and $\Lambda\leq G$ a discrete subgroup. If
$\text{rank}(G)\geq 2$ then by Theorem 8.4, $\Lambda$ is uniformly slim if and
only if it is a lattice, hence the lemma follows from the Borel density
theorem. Suppose therefore that $\text{rank}(G)=1$ and suppose that $\Lambda$
is not Zariski dense. Then $\Lambda$ is contained in a maximal proper
algebraic subgroup which is either reductive or parabolic (see [BT71]). In the
latter case pick a Langlands decomposition of the parabolic and a central
element in the Levi subgroup which acts by expansion on the unipotent radical
of the parabolic. Then by applying iterative conjugations by this element all
the elements in $\Lambda$ which do not belong to the Levi are taken to
infinity and we obtain a conjugate limit which is contained in the Levi
subgroup. Thus it is enough to deal with the case that $\Lambda\leq H$ for
some reductive subgroup $H\leq G$. Let $X=G/K$ be the symmetric space of $G$.
By a theorem of Mostow [Mo55] there is a totally geodesic subspace $Y\subset
X$ which is $H$-invariant and is a model for the symmetric space of $H$ ($Y$
is a single point in case $H$ is compact). Let $c:[0,\infty)\to X$ be any
geodesic in $X$ which starts at a point $y\in Y$ such that $\dot{c}(0)$ is
orthogonal to $Y$. Since $\text{rank}(G)=1$, $X$ is negatively curved and
hence the displacement of any $h\in H$ goes to infinity along $c$. It follows
that the injectivity radius of $\Gamma\backslash X$ goes to infinity along
$c(t)$. ∎

######  Lemma 9.15.

Let $G=G_{1}\times\cdots\times G_{n}$ be a connected semisimple Lie group and
$\Lambda\leq H$ a discrete subgroup that projects discretely to the factors.
Then $\Lambda$ admits a conjugate limit $\Delta$ which is discrete and
satisfies the following properties:

  * •

The projection of $\Delta$ to every factor is contained in the projection of
$\Lambda$ to the same factor.

  * •

The projection of $\Delta$ to a factor of $G$ is trivial unless the
intersection of $\Lambda$ with that factor is uniformly slim.

###### Proof.

Let $\Lambda_{0}=\Lambda$. For each $i=1,\ldots,n$ at its turn, if
$\Lambda_{i-1}\cap G_{i}$ is not uniformly slim then we replace
$\Lambda_{i-1}$ with a $G_{i}$-conjugate limit $\Lambda_{i}$ such that
$\Lambda_{i}\cap G_{i}=\\{1\\}$. If $\Lambda_{i-1}\cap G_{i}$ is uniformly
slim we let $\Lambda_{i}=\Lambda_{i-1}$. Then $\Delta=\Lambda_{n}$ satisfies
the desired requirements. ∎

###### Proof of Theorem 9.13.

Let $\mu$ be an ergodic component of a weak limit of
$\frac{1}{n}\sum_{i=0}^{n-1}\nu_{G}^{(i)}*\delta_{\Lambda}$, and consider the
decomposition of $G$ according to $\mu$ given by Theorem 6.5. If
$G_{\mathcal{I}}$ is nontrivial then by Proposition 7.6 and Remark 7.7 the
restriction of $\mu$ to $G_{\mathcal{I}}$ is of the form $\mu_{\Gamma}$ for
some lattice $\Gamma\leq G_{\mathcal{I}}$. In view of Corollary 9.6, $\Lambda$
intersects $G_{\mathcal{I}}$ by a lattice.

Consider now the case that $G_{\mathcal{I}}$ is trivial. Let $\Delta$ be a
generic subgroup in the support of $\mu$. Then $G$ decomposes as
$G=G_{\mathcal{H}}\times G_{\mathcal{T}}$ such that $\Delta\leq
G_{\mathcal{H}}$, all the factors of $G_{\mathcal{H}}$ are of rank one, the
projection of $\Delta$ to every factor of $G_{\mathcal{H}}$ is discrete and
the intersection of $\Delta$ with every factor of $G_{\mathcal{H}}$ is Zariski
dense. If the intersection of $\Delta$ with one of the simple factors of
$G_{\mathcal{H}}$ is a lattice then by Corollary 9.6 also $\Lambda$ intersects
this factor by a lattice. Finally we apply Lemma 9.15 to the uniformly slim
group $\Delta$ and the theorem follows. ∎

######  Remark 9.16.

The assumption that $G$ in Theorem 9.13 has property (T) is made because we do
not know if the analog of the Stück–Zimmer theorem holds for higher rank
groups of type (L). If that analog holds then one can remove the property (T)
assumption without changing the statement. Without knowing the answer to this
question one can still drop the property (T) assumption by allowing in Case
(ii) of the theorem that $H$ admits beside the rank one factors also higher
rank semisimple factors $H_{i}$ of type (L), and $\Delta\cap H_{i}$ is thin in
$H_{i}$ and uniformly slim and projects densely to the simple factors $H_{i}$.
We do not know however if such ‘irreducible’ thin uniformly slim discrete
subgroups exist.

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