Infinite Volume and Infinite Injectivity Radius
aa r X i v : . [ m a t h . G R ] F e b 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 Liegroup and let Λ ≤ G be a discrete subgroup of infinite covolume. Then, the locally symmetricspace Λ \ G/K admits injected balls of any radius. This can be considered as a geometric inter-pretation of the celebrated Margulis normal subgroup theorem. However, it applies to generaldiscrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroupsof infinite index of lattices. We establish similar results for higher rank semisimple groups withKazhdan’s property (T). We prove a stiffness result for discrete stationary random subgroups inhigher rank semisimple groups and a stationary variant of the St¨uck–Zimmer theorem for higherrank semisimple groups with property (T). We also show that a stationary limit of a measuresupported on discrete subgroups is almost surely discrete. Introduction
The statement for simple groups.
The main motivation for this paper was to prove thefollowing 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 and let X = G/K be the associated symmetric space. Let Λ ≤ G be a discrete group of infinite covolume.Then for every r > there is a point p ∈ Λ \ X where the injectivity radius is at least r . Theorem 1.1 can be regarded as a geometric interpretation of Margulis’ normal subgrouptheorem 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 Γ ≤ G be a uniform lattice and let∆ ⊳ Γ be a nontrivial normal subgroup of infinite index. The infinite volume locally symmetricspace M = ∆ \ G/K has bounded injectivity radius. Indeed, let α ∈ ∆ \ { } , let Ω be a compactfundamental domain for Γ in G/K and let D = max { d ( x, α · x ) : x ∈ Ω } . Then InjRad M ( p ) ≤ D/ p ∈ M . To see this consider a lift ˜ p ∈ G/K and let γ ∈ Γ be an element such that γ − · ˜ p ∈ Ω. Then the element γαγ − is in ∆ and has displacement at most D at ˜ p .We prove Theorem 1.1 by showing that for a certain bi- K -invariant probability measure ν = ν G on G (see §
2) the random walk on Λ \ X eventually spends most of the time in the r -thick partfor every r . Theorem 1.3.
Let M = Λ \ X be an X -orbifold of infinite volume. Let x ∈ X be an arbitrarypoint, set µ n = n P n − i =0 ν ( i ) ∗ δ x and let µ n be the pushforward of µ n to M via the covering map.For every r > and ǫ > there is N such that µ n ( M ≥ r ) ≥ − ǫ for every n ≥ N , where M ≥ r denotes the r -thick part of M and ν ( i ) the i ’th convolution of ν . Date : February 5, 2021.
Remark 1.4. ( i ) It is possible to deduce the celebrated normal subgroup theorem of Margulisby comparing Theorem 1.3 for Λ \ G with the result of Eskin and Margulis [EM02] about randomwalks on Γ \ G where Γ ≤ G is a lattice and Λ ⊳ Γ is a normal subgroup. However, our proofrelies on the St¨uck–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 alsotraces back to the factor theorem of Margulis [M78].( ii ) It follows from Theorem 1.1 that for higher rank manifolds, finite volume is equivalentto bounded injectivity radius. It might be interesting to obtain a quantitative version of thatstatement. It is also possible to deduce the result of the seven authors [7s17, Theorem 1.5] fromTheorem 1.3 by a straightforward compactness argument.1.2. Uniformly slim subgroups of semisimple groups.
More generally, let G be a connectedcentre-free semisimple Lie group. We shall say that a discrete subgroup Λ ≤ G is uniformly slim if there is a compact subset C ⊂ G such that Λ g ∩ C \ { } 6 = ∅ for every g ∈ G . In other words,Λ is uniformly slim if and only if the locally symmetric space Λ \ G/K has bounded injectivityradius. Theorem 1.1 is a special case of the following result (see Theorem 9.13 for a more generalstatement 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 . A discretesubgroup Λ ≤ G is uniformly slim if and only if there is a nontrivial normal subgroup H ⊳ G suchthat Λ ∩ H is a lattice in H . We shall say that a subgroup ∆ ≤ Γ is a conjugate limit of Λ if ∆ belongs to the closure of theconjugacy class Λ G ⊂ Sub( G ) in the Chabauty topology (see § Random walk on G/ Λ . Let G be a connected semisimple Lie group. Let Λ ⊂ G be adiscrete subgroup of G . Consider the sequence of probability measures µ n := 1 n n − X i =0 Z G δ Λ g dν ( n ) ( g ) . Every weak-* limit of the sequence µ n is a ν -stationary measure supported on the closure of theorbit of Λ in Sub( G ). We would like to know if stationary random subgroups constructed in thisway retain some properties of Λ. An essential result of this paper is that a stationary randomsubgroup is almost surely discrete (see also Theorem 2.2 for a general statement). Theorem 1.6.
Let µ ∞ be a weak-* limit of ( µ n ) . Then µ ∞ ( Sub d ( G )) = 1 where Sub d ( G ) is theChabauty open set of discrete subgroups of G . Stiffness and the St¨uck–Zimmer theorem for stationary measures.
In view of The-orem 1.6 we are led to study stationary measures supported on the space Sub d ( G ) of discretesubgroups of G , that is, ν -stationary measures µ with µ (Sub d ( G )) = 1. Relying on the re-markable theorems of Nevo and Zimmer [NZ02, NZ99] we establish that every discrete stationaryrandom subgroup of a higher rank group is (under a certain irreducibility assumption with respectto the rank one factors) an invariant random subgroup. NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 3
Theorem 1.7.
Let G be a connected centre-free semisimple Lie group without compact factorsand real rank at least two. Let µ be a ν -stationary measure on Sub d ( G ) . Suppose that µ -almostevery random subgroup intersects trivially every rank one factor of G . Then µ is invariant. Theorem 1.7 is a consequence of a decomposition theorem for stationary measures (Theorem6.5). Observe that in view of Theorem 6.5, the condition that the intersection with every rank onefactor 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 aweek 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 therecent results [BH20, BBHP20, C20].For semisimple Lie groups with Kazhdan’s property (T) we deduce the following generalizationof the St¨uck–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 and Kazhdan’s property (T). Let µ be an ergodic ν -stationary measure on Sub d ( G ) . Suppose that µ -almost every random subgroup intersects triviallyevery rank one factor of G . Then there is a semisimple factor H ⊳ G and a lattice Γ ≤ H suchthat µ = µ Γ . Here µ Γ denotes the invariant random subgroup obtained by the pushforward of the probabilitymeasure from H/ Γ to Sub( H ) ⊂ Sub( G ) via the map h Γ h Γ h − .1.5. The conclusion.
Combining Theorem 1.6, Theorem 1.9 and local rigidity we deduce thefollowing:
Theorem 1.10.
Let G be a connected centre-free semisimple Lie group without compact factors.Suppose that G has real rank at least and Kazhdan’s property (T). Let Λ ≤ G be a discretesubgroup. Suppose that for every nontrivial semisimple factor H ⊳ G the intersection Λ ∩ H isnot a lattice in H . Suppose also that no discrete conjugate limit of Λ intersects a rank one factorof G in a Zariski dense subgroup. Then n P n − i =0 ν ( i ) G ∗ δ Λ weakly converges to δ { } . See § G , not even assuming property (T).Loosely speaking, if Λ does not contain a lattice of a higher rank semisimple factor then everystationary 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-archimedeanlocal 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, wedecided to restrict to that case as well. We remark that the proof of the Nevo–Zimmer theoremalso applies with minor changes to the non-archimedean setup.
Acknowledgment.
We thank Uri Bader for sharing with us some insights concerning sta-tionary measures and Poisson boundaries. Our work was supported by the National ScienceFoundation under Grant No. DMS-1928930 while the authors participated in a program hostedby the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2020semester. The second author was partially supported by the Israel Science Foundation grant No.2919/19.
MIKOLAJ FRACZYK AND TSACHIK GELANDER Random walks on the space of discrete subgroups
Let G be a connected centre-free semisimple Lie group without compact factors and let K bea maximal compact subgroup.2.1. The measure associated to G . We let ν G be the probability measure ν G = η K ∗ δ s ∗ η K defined in [GLM, § η K denotes the normalized Haar measure on K and s ∈ G is acertain regular semisimple element with sufficiently good expanding properties when acting onthe unipotent radical of a fixed minimal parabolic subgroup (see [GLM, § ν G as the probability measure associated to G . Note that if G is semisimple, G = G × . . . × G n , wehave ν G = ν G ∗ · · · ∗ ν G n and the measures ν G i pairwise commute.2.2. The discreteness radius.
Fix a norm k · k on the Lie algebra Lie( G ) such that exp :Lie( G ) → G restricted to the unit ball B (1) = { X ∈ Lie( G ) : k X k < } is a well defineddiffeomorphism. For r ≤ B ( r ) = { X ∈ Lie( G ) : k X k < r } . For a discrete group Λ ⊂ G set I (Λ) = sup { r ≤ B ( r ) ∩ Γ = { }} . We call I (Γ) the discreteness radius of Λ.2.3. The Margulis function on the space of discrete subgroups of G . We denote bySub( G ) the space of closed subgroups of G equipped with the Chabauty topology and by Sub d ( G )the subset of discrete subgroups of G . Since G has no small subgroups, Sub d ( G ) is open (see [G18,Lemma 1.1]). We will say that a measure µ on Sub( G ) is supported on the set of discrete subgroupsif µ (Sub d ( G )) = 1. A stationary measure supported on Sub d ( G ) will be called a discrete stationaryrandom subgroup .An essential result established in [GLM] is that there is a positive constant δ = δ ( G ) such that u (Γ) := I (Γ) − δ is a Margulis function on Sub d ( G ) with respect to ν G . Theorem 2.1 ([GLM], Theorem 1.4) . There exist < c < , b ≥ such that, for every discretesubgroup Γ ≤ G , (1) Z G u (Γ g ) dν G ( g ) ≤ cu (Γ) + b. We remark that the constants δ, c and b are constructed explicitly in [GLM] in order to provecertain 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 stationarylimit of a measure supported on discrete subgroups of G is almost surely discrete. This is a keyingredient in the proofs of Theorem 1.1 and Theorem 1.3, as well as the results of § Theorem 2.2.
Let µ be a probability measure on Sub d ( G ) . Let µ n = 1 n n − X i =0 ν ( n ) G ∗ µ NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 5 and let µ ∞ be a weak-* limit of µ n . Then µ ∞ is supported on the set of discrete subgroups of G ,that is, µ ∞ is a discrete stationary random subgroup. The proof is reminiscent of [EM02].
Proof.
By restricting to compact subsets of Sub d ( G ), we may allow ourself to suppose that µ iscompactly supported and that A := R u (Λ) dµ (Λ) is finite.To prove Theorem 2.2 we need to show that(2) lim ǫ → µ ∞ ( { Λ : I (Λ) < ǫ } ) = 0 . Inequality (1) implies that Z u (Λ) d ( ν G ∗ µ )(Λ) = Z u (Λ g ) dν G ( g ) dµ (Λ) ≤ cA + b. Furthermore, iterating Condition (1) and summing the resulting geometric series we get Z u (Λ) dµ n (Λ) = Z u (Λ) d ( ν ( n ) G ∗ µ )(Λ) < c n A + C, with C := b/ (1 − c ), uniformly for all n . Set M = A + C . Then, for every τ > n ≥ µ n ( { Λ : u (Λ) ≥ M τ − } ) < τ. Setting ǫ = ( τ /M ) δ we get µ n ( { Λ : I (Λ) ≤ ǫ } ) < τ. Taking n → ∞ gives µ ∞ ( { Λ : I (Λ) ≤ ǫ } ) < τ , and letting τ → (cid:3) Essential results about discrete stationary random subgroups
In this section we assemble some results about discrete stationary random subgroups that willbe essential in the proofs of the main results. The measure ν G introduced in § K -invariant.In view of the Iwasawa decomposition, K acts transitively on G/P where P ≤ G is a minimalparabolic subgroup. Therefore there is a unique Borel regular K -invariant probability measure µ P on G/P , and the space (
G/P, µ P ) is the Poisson boundary for ( G, ν ) whenever ν is a bi- K -invariant probability measure on G . Remark 3.1.
From the universal property of the Poisson boundary we see that if ν , ν are twoprobability measures on G which correspond to the same Poisson boundary and Poisson measure( B, µ ), then they share the same stationary measures on every G -space X . Indeed, a measure µ on X is stationary with respect to ν (equivalently ν ) if and only if it is the µ -barycentre forsome measurable map from B to the space Prob( X ) of probability measures on X .The measure ν G is not smooth but obviously G admits some smooth bi- K -invariant probabilitymeasure. Therefore, in view of Remark 3.1 statements about stationary measures with respectto a measure on G which is assumed to be smooth will remain valid also for the non-smoothmeasure ν G . For simplicity we will consider, without repeating it, only bi- K -invariant probabilitymeasures ν on G although most of the statements apply to general (smooth) measures. This willallow us to make use of the special properties of ν G and in particular properties that follow from It is possible to show that some finite power ν ( n ) G is smooth, but in view of Remark 3.1 we will not need that. MIKOLAJ FRACZYK AND TSACHIK GELANDER
Inequality (1) and from Theorem 2.2. Thus in the sequel ν will refer to an arbitrary bi- K -invariantprobability measure on G while ν G is the specific measure given in § G ) the space of all discrete ν -stationary random subgroups of G . Thefollowing result from [GLM] is a consequence of Inequality (1): Theorem 3.2 ([GLM], Theorem 1.2) . The space d-SRS ( G ) is weakly uniformly discrete. Thatis, for every ǫ > there is an identity neighbourhood U ⊂ G such that for every µ ∈ d-SRS ( G ) , µ (Λ : Λ ∩ U = { } ) ≤ ǫ. Let us recall the straightforward proof when µ is a compactly supported discrete stationaryrandom subgroup. In that case Inequality (1) gives: Z u (Λ) dµ = Z u (Λ) dν ∗ µ ≤ c Z u (Λ) dµ + b. It follows that R u (Λ) dµ ≤ C := b/ (1 − c ). Thus µ ( { Λ : u (Λ) ≥ C/ǫ } ) ≤ ǫ . Thus we can take U to be the ball of radius ( ǫ/C ) δ around 1 G . Corollary 3.3 ([GLM], Corollary 1.5) . The space d-SRS ( G ) is weak-* compact.Proof. It is obvious that a limit of stationary measures is stationary and it follows from Lemma3.2 that a limit of discrete stationary random subgroups is also discrete. (cid:3)
We deduce from [BSh06, §
2] that the extreme points of the compact convex space d-SRS( G )are ergodic (see in particular [BSh06, Corollary 2.7]). Thus by the Choquet integral theoremevery discrete stationary random subgroup is a barycentre of some probability measure on the setof ergodic discrete stationary random subgroups. This fact allows us to assume ergodicity whenproving various results about discrete stationary random subgroups.Let P be a minimal parabolic subgroup of G with Langlands decomposition P = M AN . Lemma 3.4.
Let H ⊂ G be an algebraic subgroup. The homogeneous space G/H admits a ν -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 alreadyproven in [NZ02, Prop 3.2]. Our proof is different and self-contained so we present it below forcompleteness.
Proof. If H contains a conjugate of AN then G/H is compact so the existence of a ν -stationaryprobability measure is clear. Now assume that µ is a ν -stationary probability measure on G/H .Since (
G/P, µ P ) is the Poisson boundary we get a G -equivariant map κ : G/P → P ( G/H ) suchthat µ = R G/P κ ( gP ) dµ P ( gP ) . The measures κ ( gP ) are gP g − -invariant probability measuresalmost surely. By conjugating P if necessary we can assume without loss of generality that κ ( P )is a P -invariant probability measure. We will show that the existence of such a measure impliesthat H contains a conjugate of AN . Let µ be an ergodic AN -invariant component of µ . ByChevalley’s theorem [Bor, 5.1] there is a rational representation G y V and a line [ v ] ∈ P ( V )such that Stab G [ v ] = H . We will think of µ as an ergodic AN -invariant probability measure on P ( V ) supported on the G orbit of [ v ]. By Lemma 3.5, we must have µ = δ [ gv ] for some g ∈ G and [ gv ] must be fixed by AN . If follows that gHg − ⊃ AN . (cid:3) The following Lemma 3.5 is required also in the proof of Lemma 5.2.
NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 7
Lemma 3.5.
Let V be a rational real representation of AN . Then, any ergodic AN -invariantprobability measure on P ( V ) is supported on a single line fixed by AN .Proof. Let µ be an ergodic AN -invariant probability measure on P ( V ). The group N is aunipotent algebraic group so any rational representation of N has a fixed vector [Bor, 4.8]. Let V ′ be the subspace of N -fixed vectors. It is preserved by A . Since A is an R -split torus, any rationalrepresentation of A over R decomposes into a direct sum of A -eigenspaces [Bor, 8.4]. We deducethat there exists a one-dimensional subspace V ⊂ V ′ which is preserved by AN . Reasoninginductively we construct a basis e , . . . , e d of V such that V i = R e + . . . + R e i are preserved by AN and each e i is an eigenvector of A . Write χ i for the character of A such that ae i = χ i ( a ) e i . Let i be the minimal index for which µ ( P ( V i )) >
0. The sets P ( V i ) are all AN -invariant,so by ergodicity µ ( P ( V i ) \ P ( V i − )) = 1 . Consider the map ι : P ( V i ) \ P ( V i − ) → V i ι ([ x , . . . , x i − , x i ]) := ( x /x i , . . . , x i − /x i , . Let W be the space V i with the action of AN given by anw = χ i ( a ) − anw. The map P ( V i ) \ P ( V i − ) → W is AN -equivariant. The measure ι ∗ µ is an ergodic AN -invariant probabilitymeasure on W . By Lemma 3.6, ι ∗ µ = δ w for some AN invariant vector w ∈ W . This means that µ itself was supported on the line [ w + e i ], which is fixed by AN . (cid:3) 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 δ w for an AN -fixed vector w ∈ W .Proof. We prove the statement by induction, with cases dim W = 1 , W = 1 then N acts trivially because unipotent actions must fix a non-zero vector.If A acts non-trivially then µ = δ because no probability measure on R \ { } can be invariantunder dilations. Otherwise, AN acts trivially and µ = δ w for some w ∈ W , by ergodicity.Suppose dim W = 2. Let e , e be a basis of W such that e is N -invariant and both e , e areeigenvectors of A . If the action of N is nontrivial, then there exists n ∈ N and x ∈ R \ { } suchthat n ( e ) = e + x e . For every compact set K ⊂ W the intersection T k ∈ Z n k K ⊂ K ∩ ( R e ) , so every N -invariant probability measure on W must have µ ( R e ) = 1 and the lemma followsnow from the one dimensional case. Consider the case that the action of N is trivial. Thereare rational characters χ , χ of A such that ae i = χ i ( a ) e i . Let W ′ = P χ i =1 R e i . Then for anycompact subset K ⊂ W we have T a ∈ A aK ⊂ K ∩ W ′ . It follows that µ is supported on W ′ . Theaction of AN on W ′ is trivial so by ergodicity µ = δ w for some w ∈ W ′ .We move to the general case dim W ≥
3. Let W be a one dimensional subspace preserved by AN . By the inductive hypothesis, the pushforward of µ to W/W is supported on a single element.It follows that µ is supported on a single line w + W for some w ∈ W . Hence µ ( W + R w ) = 1.We have dim( W + R w ) ≤ (cid:3) Lemma 3.7.
Let Φ be a simple root system spanning a Euclidean space V . Let V ⊂ V be aproper subspace and let S = Φ ∩ V . Then (1) Φ \ S generates the whole root system Φ . (2) Φ \ S spans V .Proof. For any subset F ⊂ Φ, let h F i be the smallest subset of Φ containing F that is closedunder taking reflections. For the first assertion we need to show h Φ \ S i = Φ. Let S = S, E = V .We define inductively the subsets S i and subspaces E i . E i +1 = E i ∩ (Φ \ S i ) ⊥ , S i +1 = Φ ∩ E i +1 . MIKOLAJ FRACZYK AND TSACHIK GELANDER
By construction S i +1 ⊂ S i and E i +1 ⊂ E i . We argue that h Φ \ S i i = h Φ \ S i +1 i . The inclusion h Φ \ S i i ⊂ h Φ \ S i +1 i is clear, so it is enough to show that S i \ S i +1 ⊂ h Φ \ S i i . Let λ ∈ S i \ S i +1 .By definition, we must have λ E i +1 , so there exists a root α S i such that h λ, α i 6 = 0. Thereflection s α ( λ ) = λ − h λ, α ik α k α is not in S i because λ ∈ E i and α E i . We deduce that λ = s α ( s α ( λ )) ∈ h Φ \ S i i . This provesthat h Φ \ S i i = h Φ \ S i +1 i . The sequence of sets S i eventually stabilizes, so we have S i +1 = S i for some i . Therefore S i ⊂ (Φ \ S i ) ⊥ . The root system Φ is simple, so this is possible only if S i = ∅ . We deduce that h Φ \ S i = h Φ \ S i = . . . = h Φ \ S i i = Φ . The second assertion trivially follows from the first. (cid:3)
Remark 3.8.
The assumption that S is contained in a proper subspace is necessary in Lemma5.2. For example, the root systems B , C , G , B , C decompose as unions of two proper rootsubsystems. 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 nowon. The centralizer of A is M A . It follows that C G ( AN ) is a normal subgroup of the reductivegroup M A . Therefore it is enough to show that every semisimple element in C G ( AN ) is trivial.Let γ ∈ C G ( AN ) be a semisimple element. Let T be a maximal torus of G containing γ . Let Φbe the set of roots of T in g C . The roots naturally lie in the Euclidean space X ∗ ( T ) ⊗ R and forma root system in the classical sense. Let E ⊂ X ∗ ( T ) ⊗ R be the proper subspace spanned by theroots in m C . Then, Φ \ E corresponds to the roots in n and their opposites. Since γ commuteswith N , we must have ξ ( γ ) = 1 for every root ξ ∈ Φ \ E . On the other hand, by Lemma 3.7, theroots in Φ \ E generate Φ, so ξ ( γ ) = 1 for every root in Φ. This means that Ad( γ ) acts triviallyon g C . Since G is centre-free, γ = 1. (cid:3) Let 2 G denote the compact space of closed subsets of G equipped with the Chabauty topologyand consider the G action by conjugation. Let F ( G ) ⊂ G be the set of finite subsets of G . Lemma 3.10.
For any finite set F ⊂ G \ { } , the ν G -stationary measure lim n →∞ n n − X i =0 ν ( n ) G ∗ δ F is the Dirac measure on the empty set.Proof. It is enough to prove the assertion for F = { γ } , γ ∈ G \ { } . Moreover, by projecting toa factor we may suppose that G is simple. Let µ be the limit in question. If it is not the Diracdelta on the empty set then it gives us a ν -stationary probability measure on the closure of theconjugacy class γ G . Since γ G is a finite union of conjugacy classes, by restricting to one, say γ G ,we obtain a finite positive ν -stationary measure on G/G γ . By Lemma 3.4, up to replacing γ by a conjugate, G γ contains AN . By Lemma 3.9 we get γ = 1. Thus, we are left with showingthat µ cannot have an atom at { } . Now if µ ( { } ) = ǫ > γ and γ is unipotent. In that case by applying Theorem 2.2 to the discrete cyclic group h γ i we can choosean identity neighborhood U such that lim n →∞ n P n − i =0 ν ( n ) G ∗ δ { γ } ( U ) < ǫ , a contradiction. (cid:3) NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 9
Corollary 3.11.
The only ergodic ν -stationary probability measures on F ( G ) are δ { } and δ ∅ .Proof. In view of the dominated convergence theorem it follows from Lemma 3.10 that the only ν -stationary probability measure on G with respect to the action by conjugation is δ { } . Sincethe average of the Dirac measures on a finite set which is chosen randomly with respect to a ν -stationary measure on F ( G ) produces such a measure, the corollary follows. (cid:3) We will also make use of the following beautiful result:
Proposition 3.12 (Bader–Shalom [BSh06]) . Let H = H × H , let ν i be a probability measureon H i , i = 1 , and let ν = ν ∗ ν . Let X be an H -space and µ a ν -stationary measure on X .Then µ is ν i -stationary.Proof. In view of [BSh06, Corollary 2.7] it is enough to prove the result when µ is ergodic. For µ ergodic this is the statement of [BSh06, Lemma 3.1]. (cid:3) Lemma 3.13.
Let H = H × H be a product of centre-free semisimple Lie groups withoutcompact factors and let µ be a ν H -stationary random discrete subgroup. If the projection to H is almost surely discrete and the intersection with H is almost surely trivial, then µ is supportedon subgroups of H .Proof. Let us denote by π i the projection to H i for i = 1 ,
2. Suppose by way of contradictionthat µ is not supported on subgroups of H . Then for any sufficiently large ball B in G we have µ { Λ : ∃ α ∈ Λ with π ( α ) ∈ B, π ( α ) = 1 } > . Considering the map Λ π (Λ ∩ π − ( B )), the measure µ induces a ν H -stationary finite positivemeasure on F ( H ) which is not a combination of δ { } and δ ∅ . This contradicts Corollary 3.11. (cid:3) Remark 3.14.
Given a locally compact group G and a probability measure µ on Sub d ( G ) itis possible to construct a probability G -space ( X, m ) such that µ is the pushforward of m viathe stabilizer map X → Sub( G ) , x G x . This is proven in [7s17, Theorem 2.6] under theassumption that µ is an IRS, but the proof applies to any µ ∈ Prob(Sub( G )) which is supportedon unimodular subgroups. Thus, the study of probability measures on Sub d ( G ) is equivalent tothe 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 followsfrom the Borel density theorem for IRS [7s17, GL18] that for every ergodic discrete invariantrandom subgroup µ on G there is a decomposition G = G ′ × G ′′ such that the projection to G ′ is discrete and Zariski dense and the projection to G ′′ is trivial almost surely. The followingresults (see Theorem 4.1 and Corollary 4.4 below) generalize to invariant random subgroupsthe classical decomposition-to-irreducible-factors theorem for lattices in semisimple groups (see[Rag89, Theorem 5.22]): Theorem 4.1.
Let G = G × · · · × G n be a connected centre-free semisimple Lie group withoutcompact factors and with simple factors G i , i = 1 , . . . , n . Let µ be an ergodic discrete invariantsubgroup in G . Then G decomposes to a product of semisimple factors G = H × . . . × H k with ≤ k ≤ n , such that almost surely the projection of a random subgroup to each H i is discretewhile the projection to each proper factor of H i is dense. The proof relies on the following:
Lemma 4.2.
Let H = H × H be a product of centre-free semisimple Lie groups without compactfactors. Let µ be an ergodic discrete IRS in H which projects discretely to H and Zariski denselyto H . Then the intersection of a random subgroup with H is nontrivial almost surely.Proof of Lemma 4.2. The lemma follows immediately from Lemma 3.13. (cid:3)
Remark 4.3.
This lemma could also be proved directly (without referring to the special prop-erties of the measure ν H ) by applying disintegration of measures with respect to the factor map(Sub H , µ ) → (Sub H , π ∗ µ ). Indeed, if the intersection with H is trivial almost surely, then sincethe projection to H is Zariski dense, the H action on a generic fibre is free. By construct-ing a measurable section for the H action on a fibre one may pull the fibre measure to a left H -invariant probability measure on H . This is absurd since H is not compact. Proof of Theorem 4.1.
Let G = G ×· · ·× G n and µ be as in the statement of the theorem. We maysuppose that a random subgroup is almost surely Zariski dense. If the projection of a randomsubgroup to every proper semisimple factor of G is dense almost surely then µ is irreducible.Otherwise there is a proper decomposition G = H × H ′ such that the projection to H is almostsurely discrete. We claim that the projection to H ′ is also discrete and hence we can deducethe result by induction on the number of simple factors. Suppose by way of contradiction thatthe projection to H ′ is non-discrete and let H ⊳ H ′ be the connected component of its closure.Then H ′ decomposes as H ′ = H ′′ × H with H nontrivial. It follows that µ projects discretelyto H := H × H ′′ and densely to H . Since the intersection of every subgroup of G with H is normalized by the projection of that subgroup to H , a discrete subgroup of G that projectsdensely to H must intersect it trivially. Therefore a µ -random subgroup intersects H triviallyalmost surely. A contradiction to Lemma 4.2. (cid:3) It follows from Theorem 4.1 that an ergodic discrete IRS µ in G = G × · · · × G n is associatedwith two other IRS ˜ µ and µ given (respectively) by its projection to and the intersection withthe semisimple factors H j . Both ˜ µ and µ are products, ˜ µ = Q ˜ µ j , µ = Q µ j , where ˜ µ j and µ j are discrete IRS on H j . Obviously ˜ µ j is irreducible. We claim that also µ j is irreducible andnontrivial. If µ is irreducible then µ = ˜ µ = µ and there is nothing to prove. Otherwise, it followsfrom Lemma 4.2 that µ j is nontrivial for every j for which ˜ µ j is nontrivial. Fix j ∈ { , . . . , k } . If H j is simple then there is nothing further to prove. Suppose that H j is not simple. Let F j ⊳ H j be a proper nontrivial semisimple factor of H j . Let ∆ be a µ random subgroup and let ˜∆ j and∆ j be the corresponding random subgroups in H j , that is the projection and the intersectionwith H j . We need to show that F j ∆ j is dense in H j . Note that ∆ j ∩ F j must be trivial, sinceit is both discrete and normalized by the projection of ˜∆ j to F j , which is dense. Consider theidentity connected component of the closure of ∆ 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 ∆ j is not containedin N j . However, by construction the projection of ∆ j to H j /N j is discrete. This is absurd sincethis nontrivial discrete group is normalized by the dense projection of ˜∆ j to H j /N j . Thus wehave established the following: Corollary 4.4.
Let G = G × · · · × G n be as in Theorem 4.1. Let µ be an ergodic Zariski densediscrete invariant random subgroup in G . Let G = H × . . . × H k be the decomposition of G NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 11 to µ -irreducible factors. For every j let µ j be the IRS in H j obtain by the intersection of the µ -random subgroup with H j . Then µ j is a non-trivial irreducible discrete IRS of H j . Stationary measures of rank one groups
The celebrated factor theorem of Nevo and Zimmer [NZ02] (Theorem 6.1) is concerned withstationary 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 thefollowing 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 probabilitymeasure ν . Let ( X, µ ) be a non-trivial ergodic probability ν -stationary non-free G -system. Theneither ( X, ν ) has discrete Zariski dense stabilizers almost surely or there exists a G -equivariantmap ( X, µ ) → ( G/P, µ 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 withoutfixed points. Fix a minimal parabolic P ⊂ G with Langlands decomposition P = M AN . Everyergodic ν -stationary measure µ on the projective space P ( 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, µ P ) be the Poisson boundary. We get a G -equivariant map κ : G/P → Prob( P ( V ))such that R G/P κ ( gP ) dµ P ( gP ) = µ . Each κ ( gP ) is a gP g − -invariant measure on P ( V ). ByLemma 3.5 κ ( gP ) is supported on gAN g − fixed lines almost surely. It follows that almost allpoints in the support of µ are stabilized by a conjugate of AN . Since G/AN is compact, the G -orbits of these points are closed in P ( V ). Using ergodicity, we deduce that the measure µ mustbe supported on a single G orbit, with stabilizer conjugate to some closed subgroup H ⊃ AN . Ifthe 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. (cid:3) Lemma 5.3.
Let G be a connected simple rank one real Lie group. Let P be a minimal parabolicof G with Langlands decomposition P = M AN . Let H be closed subgroup containing AN . Theneither H ⊂ P or H = G .Proof. Let h , p , m , a , n be the Lie algebras of H, P, M, A, N respectively. Let Θ be a Cartaninvolution of G stabilizing A . Let Σ be the root system of A and let Σ + be the set of positiveroots. For any λ ∈ h ∗ let g λ = { X ∈ g | [ Y, X ] = λ ( Y ) for Y ∈ a } . We have p = g + X α ∈ Φ + g α , n = X α ∈ Φ + g α , n − = X α ∈ Φ + g − α , where n − is the Lie algebra of the unipotent radical of the opposite parabolic.First consider the case h ⊂ p . Let H be the connected component of H . We have H ⊂ P and we need to show H ⊂ P . Let h ∈ H . The set of maximal split tori in H forms a singleconjugacy class, so there exists an h ∈ H such that h := hh − normalizes A . The rank of G is one so either h commutes with A or h ah − = a − for every a ∈ A . In the first case h ∈ Z G ( A ) = M A ⊂ P . In the second h n h − = n − , which is impossible because h normalizes h . We deduce that h ∈ P and consequently h ∈ P . This proves H ⊂ P . Now consider the case h p . Since h p , there exists a positive root α ∈ Φ + such that g − α ∩ h = 0 . Let E − α ∈ g − α ∩ h be a non-zero element. For any non-zero E α ∈ R Θ( E − α ) thesubalgebra s := R E α + R E − α + R [ E α , E − α ] is isomorphic to sl ( R ) [Kn02, p. 68]. Note that since n ⊂ h , we automatically have E α ∈ h , so s is a subalgebra of h . Let Y α = [ E α , E − α ] . The rank of G is 1 so a = R Y α .We recall that the representations s ≃ sl ( R ) have symmetric weight space decomposition withrespect to a . For every ξ ∈ a ∗ we have dim( h ∩ g ξ ) = dim( h ∩ g − ξ ). Thereforedim n = dim( h ∩ n ) = X β ∈ Σ + dim( h ∩ g β ) = dim( h ∩ n − ) = dim n − , so n − ⊂ h . This proves that n + n − ⊂ h . Claim. n + n − generates g . To prove the claim we pass to the complexification g C . It will beenough to show that n C + n − C generate g C . Choose a maximal Cartan subalgebra b of m and let c = a + i b . Then, c C is a Cartan subalgebra of g C and all the roots of c in g C are real. Let Φ bethe root system of c and finally let V ⊂ c ∗ be the subspace V := { ξ ∈ c ∗ | ξ ( Y α ) = 0 } . The centralizer of a is a C + m C so the roots of c in m C are precisely those that vanish on a .Therefore n C + n − C = X ξ ∈ Φ \ V g C ,ξ . By [Kn02, Prop. 4.1.(g)], we have dim g C ,λ = 1 for every root λ ∈ Φ. Moreover, we have [Kn02,p.61] (3) [ g C ,β , g C ,γ ] = g C ,β + γ if β + γ ∈ Φ , C X β if γ = − β, X β is an element of c such that ξ ( X β ) = h ξ, β i for every ξ ∈ c ∗ . By Lemma 3.7, Φ \ V generates Φ. By (3) the set of roots in the Lie algebra generated by n C + n C is closed under takingreflections so it must be the whole root system. Using (3) once again we deduce that n C + n − C generates g C . The claim is proved.It follows that h = g , so H = G . (cid:3) Proof of Theorem 5.1.
Assume the stabilizers are not discrete almost surely. By ergodicity, thedimension of the Lie algebra of Stab G ( x ) is almost surely equal to some constant k . The actionis non-trivial so k < dim G . Let G k, g be the Grassmannian of k -planes in g and let π : G k, g ∋ W → (cid:20)^ k W (cid:21) ∈ P (cid:18)^ k g (cid:19) be the Pl¨ucker embedding. The image of the Grassmannian is a closed subset of P ( V k g ). Themap x π (lie(Stab G x)) ∈ P ( V k g ) is G -equivariant. Let µ ′ be the pushforward of µ to P ( V ).By Lemma 5.2, ( P ( V ) , µ ′ ) is supported on a single orbit G [ v ], with H := Stab G ([ v ]) ⊃ AN . Thegroup G is simple, so no k -dimensional subspace of g can be fixed by G . Since the orbit G [ v ] is These identities are true only for root systems of complex semisimple algebras, which is why we had to pass tothe complexification of g . NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 13 contained in the image of the Pl¨ucker embedding we deduce that H is a proper subgroup of G .By Lemma 5.3 H is a subgroup of P , so ( P ( V ) , µ ′ ) admits ( G/P, µ P ) as a factor.Assume now that Stab G ( x ) is not Zariski dense almost surely. The stabilizers must be infinitebecause by Lemma 3.10 there are no ν -stationary measures on the conjugacy classes of finitesubsets of G \ { } . Therefore the stabilizers must have Zariski closures of dimension between 1and dim g −
1. By ergodicity, there exists 1 ≤ k < dim g such that the Zariski closure Stab G x Z is k -dimensional almost surely. Consider the map x π (lie(Stab G x Z )) ∈ P ( V k g ). Arguing as inthe paragraph above we prove that ( X, µ ) admits (
G/P, µ P ) as a factor. (cid:3) 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 discretesubgroups. In particular we show that every such measure which is ‘irreducible’ with respectto the rank one factors of G is invariant (see Theorem 1.7). This result is a consequence of adecomposition theorem (Theorem 6.5) which extends the results of § Theorem 6.1 (Nevo-Zimmer [NZ02]) . Let G be a higher rank semisimple Lie group. Let ν be asmooth probability measure on G and let ( X, µ ) be a probability ν -stationary action of G . Theneither • µ is G -invariant, • there exists a proper parabolic subgroup Q ⊂ G and a measure preserving G -equivariantmap π : ( X, µ ) → ( G/Q, ν Q ) , where ν Q is the unique ν -stationary measure on G/Q , or • ( X, µ ) 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 andlet 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 ∈ A L be an element such that the norm of the restriction of Ad( a − ) to the Liealgebra of N is less than 1. It follows that for any two identity neighbourhoods U , U ⊂ N in N such that U is bounded there is n ∈ N such that Ad( a − n )( U ) ⊂ U .Let λ be a discrete A L -invariant random subgroup of Q . Suppose by way of contradiction that λ ( { Γ ⊂ L } ) <
1. Then there is a bounded identity neighbourhood V in Q such that λ (Ω) > { Γ : Γ ∩ V \ L = ∅} . Furthermore, we may take V to be of the form V = W · U where W ⊂ L and U ⊂ N and suppose that they are preserved by a − (e.g. we can suppose thatlog U is a norm ball in the Lie algebra of N ). Since λ is discrete we can chose a small identityneighborhood U ⊂ N so that setting V = W · U we have ǫ := λ ( { Γ ∈ Ω : Γ ∩ ( V \ L ) = ∅} ) > . Choosing n as above we get that λ (Ω a n ) ≤ λ (Ω) − ǫ in contrast with the assumption that λ is A L -invariant. (cid:3) Lemma 6.3.
Let G be a centre-free complex semisimple Lie group, let Q be a proper parabolicsubgroup. The intersection of all Levi subgroups of Q is the product of the simple factors of G contained in Q .4 MIKOLAJ FRACZYK AND TSACHIK GELANDER
Let G be a centre-free complex semisimple Lie group, let Q be a proper parabolicsubgroup. The intersection of all Levi subgroups of Q is the product of the simple factors of G contained in Q .4 MIKOLAJ FRACZYK AND TSACHIK GELANDER Proof.
We can quotient G by the product of all simple factors contained in Q . In this way we canassume that Q contains no simple factors of G . Every parabolic subgroup in G is a product ofparabolic 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 = { } . Let L be a Levi subgroup of Q and let N be the unipotent radicalof Q . Let P ⊂ Q be a Borel subgroup with a Levi subgroup A ⊂ L . Then A is a maximal torusof G . Write g , q , l , n , p , a for the corresponding Lie algebras. Let Φ be the root system of g withrespect to a . Let Φ + be the set of positive roots corresponding to P and let Π ⊂ Φ + be the subsetof simple roots. Let W be the Weyl group. Finally let F ⊂ Π be the subset of simple roots lyingin q . Write E ⊂ a ∗ for the space spanned by F and let S = Φ ∩ E . Then l = X λ ∈ S g λ , n = X λ ∈ Φ + \ S g λ , q = l + n = X λ ∈ S ∪ Φ + g λ . Let R := Φ + \ S . By Lemma 3.7, R spans a ∗ . Let A ′ ⊂ A be a maximal torus of J . Since both J and N are normal in Q we have [ J, N ] ⊂ J ∩ N ⊂ L ∩ N = { } . The torus A ′ commutes with N so the roots in R vanish on its Lie algebra a ′ . As R spans a ∗ we get a ′ = 0. As a normal subgroupof L , J must be reductive, so triviality of maximal tori implies that J is finite. Since J is finiteand normal in P it must be in the centre, so J = { } . (cid:3) Lemma 6.4.
Let G be a centre-free semisimple Lie group with proper parabolic subgroup P ⊂ Q .Let Λ ⊂ Q be a discrete P -invariant random subgroup of Q . Then there is a proper semisimplefactor H of G such that Λ is supported on discrete subgroups of H almost surely.Proof. Let L ≤ Q be a Levi subgroup of Q , such that the centre A of L is contained in P .By Lemma 6.2, Λ ⊂ L almost surely. Since Λ is P -invariant we can upgrade this inclusion toΛ ⊂ T p ∈ P L p = T q ∈ Q L q . Let J be the intersection of all Levi subgroups of Q . The set ofreal points of Q is Zariski dense so J coincides with the real points of the intersection of allcomplex Levi subgroups of Q ( C ). By Lemma 6.3 we deduce that Λ is supported on Sub d ( H )where H = T q ∈ Q L q is a proper semisimple factor of G almost surely. (cid:3) The following decomposition theorem generalizes Theorem 4.1 from invariant random sub-groups to stationary discrete subgroups:
Theorem 6.5.
Let G be a connected centre-free semisimple Lie group without compact factorsand µ a discrete ν -stationary random subgroup of G . Then G decomposes to a product of threesemisimple factors G = G I × G H × G T such that (1) µ projects to an IRS in G I for which all the irreducible factors are of rank at least . (2) G H is a product of rank one factors and µ projects discretely to every factor of G H . (3) µ projects trivially to G T .Furthermore, the intersection of a random subgroup with every simple factor of G H as well aswith every irreducible factor of G I is almost surely Zariski dense in that factor. By the irreducible factors of G I we mean the irreducible factors associated with the decom-position of IRS as in Theorem 4.1. The subscripts I , H , T stands for invariant, hyperbolic andtrivial (respectively). NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 15
Proof of Theorem 6.5.
We will argue by induction on the number of simple factors of G . If G issimple then we may suppose rank( G ) ≥ µ 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 µ is not invariant. Let π : (Sub( G ) , µ ) → ( G/Q, µ Q ) be the measure preserving G -equivariant map afforded by Theorem6.1. Let P ⊂ Q be a minimal parabolic.Let P (Sub( G )) be the space of Borel probability measures on Sub( G ). As a first step we willconstruct a G -equivariant map κ : ( G/P, µ P ) → P (Sub( G )) , such that µ = R G/P κ ( gP ) dµ P ( gP ) . Consider the map ψ : G → P (Sub( G )) given by ψ ( g ) := g ∗ µ .Then, ψ is a G -equivariant, bounded ν -harmonic function on G . The pair ( G/P, µ P ) is theFurstenberg–Poisson boundary, so there exists a unique measurable function κ : ( G/P, µ P ) →P (Sub( G )) such that µ = R G/P κ ( gP ) dµ P ( gP ) . The uniqueness implies that the map is G -equivariant. This also implies that κ ( gP ) is gP g − invariant µ P -almost surely.Consider the composition π ∗ ◦ κ : ( G/P, µ p ) → P ( G/Q ) . This map is G -equivariant so themeasure π ∗ ◦ κ ( gP ) is gP g − -invariant almost surely. The unique probability measure on G/Q with this property is δ gQ . Hence π ◦ κ ( gP ) = gQ almost surely.By comparing the stabilizers we deduce that for µ P almost every gP ∈ G/P , the measure κ ( gP )is supported on the set of discrete subgroups of gQg − . The measure κ ( gP ) is gP g − -invariant,so it is the distribution of a gP g − -invariant random discrete subgroup of gQg − . By Lemma 6.4we get κ ( gP ) = δ { } almost surely. Since µ = R G/P κ ( gP ) dµ P ( gP ) we deduce that µ = δ { } , acontradiction. This completes the proof of the theorem when G is simple.Suppose now that G = Q ni =1 G i has n > ν = ν ∗ · · · ∗ ν n where ν i is a probability measure on G i . If µ is invariant the theorem followsfrom Theorem 4.1 and Corollary 4.4. If µ admits a parabolic factor ( G/Q, µ Q ) then arguing asin the simple case above we deduce from Lemma 6.4 that µ is supported on discrete subgroupsof H for some proper semisimple factor H ⊳ G . Then we deduce the result from the inductionhypothesis.Thus we are left with the case where µ 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 , theprojection of a µ -random subgroup is almost surely discrete and Zariski dense in G . Considerthe intersection with H = G × . . . × G n . Since the intersection is a ν ∗ · · · ∗ ν n stationary randomsubgroup it follows from the induction hypothesis that there is a decomposition H = H I × H H × H T as in the statement of the theorem. Let ∆ denote the µ -random subgroup. Since the intersection∆ ∩ ( H I × H H ) is almost surely Zariski dense in H I × H H and normalized by the projection of ∆to H I × H H , it follows that this projection is discrete as well. Thus the projection of a µ -randomsubgroup to G × H I × H H is discrete almost surely. Applying Lemma 3.13 we deduce thata random subgroup projects trivially to H T . Furthermore, by the same reasoning we see thatthe projection of ∆ to every irreducible factor of H I as well as to every simple factor of H H isdiscrete. Finally, consider the intersection of the random subgroup with G . By Lemma 3.13 thisintersection is non-trivial and since it is normalized by the projection to G which by assumptionis Zariski dense we deduce that the intersection with G is also Zariski dense. Thus our desired decomposition is given by G I = H I , G H = G × H H , G T = H T . (cid:3) We are now in a position to deduce:
Theorem 6.6 (Theorem 1.7 of the introduction) . Let G be a connected centre-free semisimpleLie group without compact factors and real rank at least two. Let µ be a ν -stationary measure onSub d ( G ) . Suppose that µ -almost every random subgroup intersects trivially every rank one factorof G . Then µ is invariant.Proof of Theorem 1.7. The assumption on the rank one factors implies that in the decompositionof G according to µ given by Theorem 6.5 the factor G H is trivial. (cid:3) Remark 6.7.
The assumption that for every rank one factor the intersection is trivial can bereplaced 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 con-struction demonstrates how to produce a stationary non-invariant measure on the space of discretesubgroups: Example 6.8.
Let T be a 3 regular tree and consider X to be its grandfather graph accordingto a chosen point at the visual boundary ∂T . That is, X is obtained from T by adding edgesbetween 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 hence8 colorable using 3 colors for T and another 5 for the new edges. By doubling every edge andgiving it the two possible orientations we obtain a Schreier graph F /H associated with some‘uniformly slim’ subgroup H ≤ F . Note that the closure of the conjugacy class C of H in F 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 -invariantprobability measure on C . Indeed F acts transitively on T and hence preserves no probabilitymeasure on ∂ ( T ). Embed F as a lattice in SL(2 , R ) and let Λ be the image of H . Then everyweak- ∗ limit of the random walk n P n − i =0 ν ( i ) G ∗ δ Λ is a stationary measure on Sub d (SL(2 , R )) whichin view of [NZ99, Lemma 6.1] cannot be invariant. Similar constructions can be made for everyrank one group that admits a lattice which projects on F .7. St¨uck–Zimmer theorem for stationary measures
The decomposition result, Theorem 6.5, allows to extend the celebrated St¨uck–Zimmer theoremto stationary measure with discrete stabilizers.
Theorem 7.1 (Theorem 1.9 of the introduction) . Let G be a connected centre-free semisimpleLie group without compact factors. Suppose that G has real rank at least and Kazhdan’s property(T). Let µ be an ergodic ν -stationary measure on Sub d ( G ) . Suppose that µ -almost every random Nevo and Zimmer constructed stationary actions of rank one groups on compact spaces which admit no invariantprobability measure [NZ99, Theorem B]. However in their construction, the pushforward of every stationary measureto the space of discrete subgroups via the stabilizer map collapses to an invariant measure on Sub d ( G ). NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 17 subgroup intersects trivially every rank one factor of G . Then there is a semisimple factor H ⊳ G and a lattice Γ ≤ H such that µ = µ Γ . We will rely on the following proper ergodicity result:
Lemma 7.2.
Let G = G × G where G , G are locally compact second countable groups. Let X be a compact, second countable G -space with G -invariant probability measure µ such that Stab G ( x ) is discrete and has a dense projection onto G for µ -almost every x ∈ X . Then µ is G -invariant.Proof. In view of the ergodic decomposition of probability measure preserving actions, the lemmaclearly reduces to the ergodic case, so let us assume that µ is G -ergodic. Choose a smoothsymmetric probability measure η on G whose support generates G . Write η g for the measure η g ( A ) = η ( g − Ag ). Since the support of η generates G , the measure µ is ergodic with respectto the random walk on X induced by η g , for every g ∈ G . We say that a point x ∈ X is η g -generic for µ if(4) lim n →∞ n n − X i =1 Z f ( gx ) d ( η g ) ∗ i ( g ) = lim n →∞ n n − X i =1 Z f ( g − gg x ) dη ∗ i ( g ) = Z f dµ ( x ) , for all continuous functions f . By the pointwise ergodic theorem for random walks we know thatthe set of η g -generic points has full measure with respect to µ . It follows that the set { ( g , x ) | x is not g ηg − -generic } has zero measure. By Fubini’s theorem we deduce that there is a subset X ′ ⊂ X with µ ( X ′ ) = 1,such that every x ∈ X ′ satisfies the following property. For almost every g ∈ G (5) lim n →∞ n n − X i =1 ( η g ) ∗ i δ x = µ. in the weak-* topology.Take a point x ∈ X ′ such that Γ := Stab G ( x ) is discrete and has a dense projection onto G .Since Γ is countable, we may fix g ∈ G such that (5) holds for g γ , for every γ = ( γ , γ ) ∈ Γ.Note that η γg = η γ g . Comparing (5) for g and γ g and using the fact that γ − x = x we findthat µ = lim n →∞ n n − X i =1 ( η γ g ) ∗ i δ x = lim n →∞ n n − X i =1 ( η γg ) ∗ i δ x = lim n →∞ n n − X i =1 γ ∗ ( η g ) ∗ i δ γx = γ ∗ lim n →∞ n n − X i =1 ( η g ) ∗ i δ x = γ ∗ µ. This means that Γ G ⊂ Stab G µ . The action of G on the set of probability measures on X iscontinuous, so Stab G µ ⊃ Γ G = G. (cid:3) Corollary 7.3.
Let G and G be locally compact second countable groups. Let µ be an ergodicdiscrete invariant random subgroup in G × G with dense projections to G . Then µ is G -ergodic.Proof. Let µ be an ergodic discrete invariant random subgroup in G × G with dense projectionsto G . In view of [7s17, Theorem 2.6] µ can be realised as the stabilizer of some compact, second countable G -space X . Abusing notations we will denote the measure on X by µ as well. Considerthe ergodic decomposition of ( X, µ ) with respect to the G -action. By Lemma 7.2 the G -ergodiccomponents of µ are G -invariant. Since µ is G × G -ergodic it follows that the decompositionis trivial. This means that µ is G -ergodic. (cid:3) Thus we have established:
Corollary 7.4.
Under the assumptions of Corollary 4.4, the IRS µ j are properly ergodic, that is, µ j is ergodic with respect to every simple factor of H j . Recall the main result of [SZ94] (see also [7s17, § Theorem 7.5 (St¨uck–Zimmer [SZ94]) . Let G be a centre-free semisimple Lie group of real rankat least 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 anormal subgroup N ⊳ G and a lattice Γ < N such that for almost every x ∈ X the stabilizer of x is conjugate to Γ . The following is a straightforward conclusion of the combination of Theorem 4.1, Corollary 4.4,Corollary 7.4, and the St¨uck Zimmer theorem:
Proposition 7.6.
Let G = G × · · · × G n be a connected centre-free semisimple Lie group withoutcompact factors. Suppose that G has Kazhdan’s property (T). Let µ be an ergodic Zariski densediscrete invariant subgroup in G , and suppose that the projection of a µ -random subgroup to everyrank one simple factor of G is non-discrete. Then µ is supported on lattices in G . Remark 7.7.
Note that in the context of Proposition 7.6 the three invariant random subgroups µ, ˜ µ and µ 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 µ 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 µ is not δ { } . As in the proofof Theorem 1.7 we get that G = G I × G T . Since all the irreducible factors of G I are of rank atleast 2, the result follows from Proposition 7.6. (cid:3) Margulis conjecture for simple Lie groups of rank at least . 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 ν be a smooth probabilitymeasure on G . Let µ be an ergodic ν -stationary random discrete subgroup of G . Then µ = δ { } ,or there exists a lattice Γ ⊂ G such that µ = µ Γ . This allows us to prove:
Theorem 8.2.
Let G be a centre-free higher rank simple Lie group. Let Λ ≤ G is a discretesubgroup of infinite covolume. Then n n − X i =0 ν ( i ) G ∗ δ Λ → δ { } . NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 19
Proof.
Let µ be a stationary limit of n P n − i =0 ν ( i ) G ∗ δ Λ . Then µ is supported on the closure of theconjugacy class of Λ in Sub( G ). Moreover, in view of Theorem 2.2, µ (Sub d ( G )) = 1. Since G issimple of higher rank, lattices in G are locally rigid. Thus the closure of the conjugacy class of Λin Sub( G ) contains no lattices. It follows from Theorem 8.1 that µ = δ { } . (cid:3) Theorem 8.2 can be reformulated as follows: Let M = Λ \ G/K be a locally symmetric space ofinfinite volume, where G is a simple Lie groups of rank at least 2. Consider the ν G -random walkon M starting at some point x . Then for every r >
0, the random walk will almost certainlyeventually spend most of the time in the r -thick part. More precisely, let ˜ x ∈ G/K be a lift of x and let m n ∈ Prob( M ) be the pushforward of n P n − i =0 ν ( i ) G ∗ δ ˜ x via the covering map. Thenwe have: Corollary 8.3.
For every r > and ǫ > there is N such that, for n ≥ N , m n ( M ≥ r ) > − ǫ . (cid:3) 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 ∆ ≤ G be a discretesubgroup. Then ∆ is uniformly slim if and only if ∆ is a lattice in G .Proof. Latices are obviously uniformly slim. On the other hand if ∆ ≤ G is of infinite covolume,it follows from Theorem 8.2 that { } is a conjugate limit of ∆. Thus ∆ is not uniformly slim. (cid:3) Uniformly slim subgroups of semisimple groups
Throughout this section we will assume that G = G × · · · × G m is a connected centre-freesemisimple Lie group without compact factors. The G i ’s are the simple factors of G and wesuppose that m ≥
2. Since the normal subgroup theorem requires higher rank and the St¨uck–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 rankone. Among them Sp( n, , n ≥ F − have Kazhdan’s property (T) while the real andcomplex hyperbolic groups PO( n,
1) and PU( n, , n ≥ Definition 9.1.
We shall say that a semisimple group G is of type (L) if every simple factor of G is either PO( n,
1) or PU( n, , n ≥ Theorem 9.2.
Let G be a connected centre-free semisimple Lie group without compact factors.Let Λ be a discrete subgroup of G and suppose that for every nontrivial normal subgroup H ⊳ G ,the intersection Λ ∩ H is not a lattice in H . Let µ be an ergodic component of a stationary limitof n P n − i =0 ν ( i ) G ∗ δ Λ . Then there are (possibly trivial) normal subgroups H , H ⊳ G , where H isa product of rank one factors, H is of type (L) and H ∩ H = { } , such that µ -almost surely: (1) the random group is contained in H × H and projects discretely to H and to H , (2) the projection to H is an IRS whose irreducible factors are higher rank and the intersec-tion with each factor is thin (i.e. Zariski dense of infinite co-volume in the factor), (3) the intersection with every simple factor of H is Zariski dense, and (4) the intersection with every simple factor of H with property (T) is thin. Remark 9.3.
It is a famous open problem whether the analog of the St¨uck–Zimmer theoremholds for higher rank groups of type (L). A positive solution to this problem would imply that thegroup H in Theorem 9.2 is trivial. On the other hand the group H may not be trivial even if Λ is not uniformly slim. Using ideas from [7s20] one can construct a non-uniformly slim subgroupof 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 instanceif Λ does not contain large commuting subgroups then so does a random group with respect toa stationary limit. Such data eliminates some of the possibilities that the general case allows. Inthe special case that Λ 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 δ { } . Lemma 9.5.
Let H ⊳ G be a normal subgroup of G and let Γ ≤ H be a lattice. Suppose thateither rank ( H ) ≥ and Γ is irreducible or that H has Kazhdan’s property (T). Then Γ is locallyrigid in G .Proof. The case where rank( H ) ≥ H has property (T). Let f : Γ → G be a deformation andconsider the unitary representation of Γ on L ( G/H ) where Γ acts via f . If f is a sufficiently smalldeformation this representation has a (Σ , ǫ )-invariant unit vector where Σ is a finite generatingset for Γ and ǫ = ǫ (Σ) > f (Γ) liesin a compact subgroup of G/H . By further conjugating by some element close to the identity wemay assume that it lies in a fixed compact subgroup K ≤ G/H . By [Wa75, Theorem 2.6] Γ hasonly finitely many conjugacy classes of representations into K . Thus we deduce that eventually f (Γ) ≤ H . The result now follows from local rigidity of Γ in H . (cid:3) Corollary 9.6.
Let H ⊳ G be as in Lemma 9.5. Let Λ be a discrete subgroup of G . Suppose that Λ admits a discrete conjugate limit which intersects H by an irreducible lattice. Then Λ intersects H by a lattice.Proof. Let ∆ be a discrete conjugate limit of Λ and suppose that Γ = ∆ ∩ H is a lattice in H .Fixing a finite presentation h Σ : R i for Γ we see that if Λ g is sufficiently close to ∆ then the mapsending each σ ∈ Σ to its nearest element in Λ g extends to a homomorphism f g : Γ → Λ g . Thusthe corollary follows from Lemma 9.5. (cid:3) Proof of Theorem 9.2.
By Theorem 2.2, µ is a discrete stationary random subgroup. Considerthe decomposition of G given by Theorem 6.5, G = G I × G H × G T and set H = G H , H = G I .By Theorem 6.5, H is a product of rank one factors. Suppose by way of contradiction that H is not of type (L). Then by Proposition 7.6, Remark 7.7 and Remark 7.8 a random subgroupintersects a higher rank semisimple factor of G I by an irreducible lattice. This is also the case ifItem (2) of the theorem is not satisfied. Similarly, if Item (4) of the theorem is not satisfied thena 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 Λ itself intersects the given factor by alattice. This is a contradiction to the assumption. (cid:3) We deduce the following classification of uniformly slim subgroups when all the factors of G are higher rank: Theorem 9.7.
Let G = G × · · · × G n be a connected centre-free semisimple Lie group such thatrank ( G i ) ≥ , i = 1 , . . . , n . Let Λ be a discrete subgroup. Then Λ is uniformly slim if and onlyif there is a nontrivial semisimple factor H ⊳ G such that Λ ∩ H is a lattice in H . If Λ is notuniformly slim then n P n − i =0 ν ( i ) G ∗ δ Λ → δ { } . NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 21
Definition 9.8.
Let G be a locally compact group. A subset D ⊂ G × G will be called a densitytester if it satisfies the following property: A subgroup Γ ≤ G is dense if and only if (Γ × Γ) ∩ D = ∅ .Note that any connected semisimple Lie group G admits a compact density tester. Indeed, wecan 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 ∈ U, v ∈ V , we have that log( u ) andlog( v ) generate the Lie algebra of G . Then it follows that for any such pair the group h u, v i isdense in G (see [BG03] or [GZ02] for details). Thus D = U × V is a density tester. Definition 9.9.
Let G = G × · · · × G n be a connected semisimple Lie group with simple factors G i . Let Γ ≤ G be a discrete subgroup. We will say that Γ is uniformly irreducible if there are: • a compact density tester D i for G/G i for i = 1 , . . . , n • a compact set B ⊂ G such that for every g ∈ G and i , the projection of the finite set Γ g ∩ B to G/G i admits two points γ i , γ i such that ( γ i , γ i ) ∈ D i . Remark 9.10.
Obviously a uniformly irreducible subgroup is uniformly slim. It is easy to checkthat every irreducible lattice in G is uniformly irreducible. It is also not hard to check that anontrivial normal subgroup of a uniform irreducible lattice is uniformly irreducible. Theorem 9.11.
Let G = G × · · · × G n , n ≥ be a connected centre-free semisimple Lie groupwithout compact factors and suppose that at least one of the simple factors of G has Kazhdan’sproperty (T). Let Λ be a discrete subgroup of G . Then Λ is uniformly irreducible if and only if Λ is an irreducible lattice in G .Proof. Let Λ ≤ G be a uniformly irreducible discrete subgroup. Note that every conjugatelimit of a uniformly irreducible group is uniformly irreducible. In view of that, it follows fromTheorem 2.2, Proposition 7.6 and Remark 7.8 that every ergodic component of a stationary limitof n P n − i =0 ν ( i ) G ∗ δ Λ , must be µ Γ for some irreducible lattice in G . In view of Corollary 9.6 thisimplies that Λ is an irreducible lattice in G . (cid:3) Let G be as above and let G i be a simple factor. We shall say that a discrete subgroup Λprojects uniformly densely to G i if there is a compact set B ⊂ G and a compact density tester D for G i such that for every g ∈ G the projection of Λ g ∩ B to G i contains two points α, β suchthat ( α, β ) ∈ D . The following is a variant of Theorem 9.11. We leave the proof as an exercise. Proposition 9.12.
Let G = G × · · · × G n , n ≥ be a connected centre-free semisimple Liegroup without compact factors and with Kazhdan’s property (T). Let Λ be a discrete uniformlyslim subgroup of G . Suppose in addition that Λ projects uniformly densely to every rank onefactor of G . Then there is a nontrivial semisimple factor H ⊳ G which contains all the rank onefactors of G such that Λ ∩ H is a lattice in H . We shall now prove the following alternative:
Theorem 9.13.
Let G = G × · · · × G n , n ≥ be a connected centre-free semisimple Lie groupwithout compact factors and with Kazhdan’s property (T). Let Λ ≤ G be a discrete uniformly slimgroup. Then exactly one of the following holds: (i) There is a nontrivial proper semisimple factor H ⊳ G such that Λ ∩ H is a lattice in H . (ii) There is a proper nontrivial semisimple factor H ⊳ G which is a product of rank one simplefactors, H = Q ki = j G i j , and a conjugate limit ∆ of Λ , such that ∆ is a discrete uniformlyslim subgroup of H . Furthermore ∆ intersects every simple factor of H in a uniformlyslim 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 Λ ≤ G a discrete subgroup. If rank( G ) ≥ G ) = 1 and suppose that Λ is not Zariski dense. Then Λis 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 elementin the Levi subgroup which acts by expansion on the unipotent radical of the parabolic. Then byapplying iterative conjugations by this element all the elements in Λ which do not belong to theLevi 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 Λ ≤ H for some reductive subgroup H ≤ G . Let X = G/K be the symmetric space of G . By a theorem of Mostow [Mo55] there is a totallygeodesic subspace Y ⊂ 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 , ∞ ) → X be any geodesic in X which startsat a point y ∈ Y such that ˙ c (0) is orthogonal to Y . Since rank( G ) = 1, X is negatively curvedand hence the displacement of any h ∈ H goes to infinity along c . It follows that the injectivityradius of Γ \ X goes to infinity along c ( t ). (cid:3) Lemma 9.15.
Let G = G × · · · × G n be a connected semisimple Lie group and Λ ≤ H a discretesubgroup that projects discretely to the factors. Then Λ admits a conjugate limit ∆ which isdiscrete and satisfies the following properties: • The projection of ∆ to every factor is contained in the projection of Λ to the same factor. • The projection of ∆ to a factor of G is trivial unless the intersection of Λ with that factoris uniformly slim.Proof. Let Λ = Λ. For each i = 1 , . . . , n at its turn, if Λ i − ∩ G i is not uniformly slim then wereplace Λ i − with a G i -conjugate limit Λ i such that Λ i ∩ G i = { } . If Λ i − ∩ G i is uniformly slimwe let Λ i = Λ i − . Then ∆ = Λ n satisfies the desired requirements. (cid:3) Proof of Theorem 9.13.
Let µ be an ergodic component of a weak limit of n P n − i =0 ν ( i ) G ∗ δ Λ , andconsider the decomposition of G according to µ given by Theorem 6.5. If G I is nontrivial thenby Proposition 7.6 and Remark 7.7 the restriction of µ to G I is of the form µ Γ for some latticeΓ ≤ G I . In view of Corollary 9.6, Λ intersects G I by a lattice.Consider now the case that G I is trivial. Let ∆ be a generic subgroup in the support of µ .Then G decomposes as G = G H × G T such that ∆ ≤ G H , all the factors of G H are of rank one,the projection of ∆ to every factor of G H is discrete and the intersection of ∆ with every factorof G H is Zariski dense. If the intersection of ∆ with one of the simple factors of G H is a latticethen by Corollary 9.6 also Λ intersects this factor by a lattice. Finally we apply Lemma 9.15 tothe uniformly slim group ∆ and the theorem follows. (cid:3) Remark 9.16.
The assumption that G in Theorem 9.13 has property (T) is made because wedo not know if the analog of the St¨uck–Zimmer theorem holds for higher rank groups of type(L). If that analog holds then one can remove the property (T) assumption without changing NFINITE VOLUME AND INFINITE INJECTIVITY RADIUS 23 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 factorsalso higher rank semisimple factors H i of type (L), and ∆ ∩ H i is thin in H i and uniformly slimand projects densely to the simple factors H i . We do not know however if such ‘irreducible’ thinuniformly slim discrete subgroups exist. References [7s17] M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I. Samet, On the growth of L -invariants for sequences of lattices in Lie groups, Annals of Math. 185 (2017), 711–790.[7s20] M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I. Samet, On the growth ofL2-invariants of locally symmetric spaces, II: exotic invariant random subgroups in rank one. Int. Math.Res. Not. IMRN 2020, no. 9, 2588–2625.[BSh06] U. Bader, Y. Shalom, Factor and normal subgroup theorems for lattices in products of groups. Invent.Math. 163 (2006), no. 2, 415–454.[BBHP20] U. Bader, R. Boutonnet, C. Houdayer, J. Peterson, Charmenability of arithmetic groups of producttype, arXiv:2009.09952.[BH20] R. Boutonnet, C. Houdayer, Stationary characters on lattices of semisimple Lie groups, arXiv:1908.07812.[BG03] E. Breuillard, T. Gelander, On dense free subgroups of Lie groups. J. Algebra 261 (2003), no. 2, 448–467.[Bor] A. Borel. Linear algebraic groups. Vol. 126. Graduate Texts in Mathematics Springer, 1991.[BT71] A. Borel and J. Tits. ´El´ements unipotents et sous-groupes paraboliques de groupes r´eductifs. I. Invent.Math., 12:95–104, 1971.[C20] D. Creutz, Stabilizers of Stationary Actions of Lattices in Semisimple Groups, arXiv:2010.13987.[LG19] I. Gekhtman, A. Levit, Critical exponents of invariant random subgroups in negative curvature. Geom.Funct. Anal. 29, (2019) 411—439.[G18] T. Gelander, Kazhdan-Margulis theorem for invariant random subgroups. Adv. Math. 327 (2018), 47–51.[GL18] T. Gelander, A. Levit, Invariant random subgroups over non-Archimedean local fields. Math. Ann. 372(2018), no. 3-4, 1503–1544.[GLM] T. Gelander, A. Levit, G.A. Margulis, Effective discreteness radius of stabilisers for stationary actions,preprint.[GZ02] T. Gelander, A. ˙Zuk, Dependence of Kazhdan constants on generating subsets. Israel J. Math. 129 (2002),93–98.[HT16] Y. Hartman, O. Tamuz, Stabilizer rigidity in irreducible group actions. Israel J. Math. 216 (2016), no. 2,679–705.[HM79] R. Howe, C. C. Moore. ”Asymptotic properties of unitary representations.” J. Func-tional Anal. 32 (1979),72–96.[M90] G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, 1990.[M78] G.A. Margulis, Factor groups of discrete subgroups and measure theory. (Russian) Funktsional. Anal. iPrilozhen. 12 (1978), no. 4, 64–76.[Kn02] A. W. Knapp, Lie groups beyond an introduction. Second edition. Progress in Mathematics, 140.Birkh¨auser Boston, Inc., Boston, MA, 2002.[NZ99] A. Nevo, Amos, R.J. Zimmer, Homogenous projective factors for actions of semi-simple Lie groups. Invent.Math. 138 (1999), no. 2, 229–252.[NZ02] A. Nevo, Amos, R.J. Zimmer, A structure theorem for actions of semisimple Lie groups. Ann. of Math.(2) 156 (2002), no. 2, 565–594.[NZ02b] A. Nevo, Amos, R.J. Zimmer, A generalization of the intermediate factors theorem. J. Anal. Math. 86(2002), 93–104.[SZ94] G. St¨uck, R. J. Zimmer. ”Stabilizers for Ergodic Actions of Higher Rank Semisimple Groups.” Annals ofMath. 139 (1994), 723–47.[EM02] A. Eskin, G. Margulis. ”Recurrence properties of random walks on finite volume homogeneous manifolds.”Random walks and geometry (2002), 431–444.4 MIKOLAJ FRACZYK AND TSACHIK GELANDER