Stabilizers of Stationary Actions of Lattices in Semisimple Groups

Every stationary action of a strongly irreducible lattice or commensurator of such a latiice in a general semisimple group, with at least one higher-rank connected factor, either has finite stabilizers almost surely or finite index stabilizers almost surely.


Introduction
Nevo-Stuck-Zimmer [SZ94], [NZ99a], [NZ99b] proved that every ergodic measure-preserving action of a lattice in a connected semisimple Lie group, all of whose simple factors are higher-rank, either has finite stabilizers almost everywhere or finite index stabilizers almost everywhere.
The author and Peterson [CP17] extended this to lattices in general semisimple groups, and the author [Cre17] relaxed the higher-rank requirement to a single factor.However, for nonamenable groups such as lattices, measure-preserving actions is not the correct context as actions on compact metric spaces will not, in general, admit invariant measures.
The natural setting for the study of actions of lattices is that of stationary actions (as stationary measures do always exist).There is a natural measure on a lattice (related to the Poisson boundary of the ambient group) and ww say an action of a lattice is stationary when it is stationary with respect to that measure (see Definition 1.2).Corollary 6.5 states: Theorem.Let Γ < G be a strongly irreducible lattice in a semisimple group with finite center and no compact factors, at least one simple factor of higher-rank.(Strong irreducibility meaning the projection of the lattice is dense in every proper subproduct of G).

Every ergodic stationary action of Γ has finite stabilizers almost everywhere or finite index stabilizers almost everywhere.
We also obtain the same dichotomy for dense commensurators of lattices (Corollary 6.4).
As every action on a compact metric space admits a stationary measure we obtain: every action of Γ on a minimal compact metric space is either topologically free or the space is finite.Consequently, every ergodic uniformly recurrent subgroup is finite (see [GW14] for the notion of uniformly recurrent subgroups), answering the generalized form of the question posed in [GW14] as Problem 5.4.
Omitting the requirement of a higher-rank factor but requiring at least two factors, every stationary action either has finite stabilizers or is orbit equivalent to an action of Z.

Stationary Intermediate Factor Theorems and Induced Actions
We establish an intermediate factor theorem along the lines of those of Stuck-Zimmer [SZ94] for stationary actions (Theorem 2.3): every intermediate factor between the stationary space and the stationary join of the space with the Poisson boundary (which is the product space in the measure-preserving case) must have certain structure.We generalize the induced action [SZ94] to the stationary setting (Definition 3.1) to apply the factor theorem to actions of lattices.Similarly, we generalize the intermediate factor theorem for commensurators of [CP17] to the stationary case: Theorem A.2.

Projecting Actions to the Ambient Group
The other main tool we introduce is a technique for projecting the action of a lattice or commensurator to the ambient group: Theorems 4.3 and 4.1.These techniques give a very general method for relating the stabilizers of the action of a lattice or commensurator to the stabilizers of an action of the ambient group (on some other space).The technique is summarized as: if Λ is a dense commensurator of a lattice in a locally compact second countable group G and η is a stationary random subgroup of Λ then there exists a stationary random subgroup of G via the closure map on subgroups, see Corollary 4.2.

Related Work
Boutonnet and Houdayer [BH19] proved an operator-algebraic statement that all stationary characters on a lattice in a connected semisimple group, all simple factors of higher-rank, are trivial or arise from finite-index subrepresentations; in particular their result implies that stationary actions of such lattices have finite stabilizers or finite index stabilizers.Our result covers a much larger class of lattices, including those where only one simple factor is higher-rank and those with p-adic components.
Extremely recently, [BRHP20], Bader, Boutonnet, Houdayer and Peterson, independently obtained the result on stationary actions of general lattices as a consequence of an incredibly strong operatoralgebraic rigidity statement for lattices.Our result on stationary actions of lattices in products of groups, announced prior to their work (e.g.[Cre18]), relies on dynamical methods very different from their algebraic methods.

Stationary Actions of Lattices
a lattice in a product of locally compact second countable groups.
Γ is strongly irreducible when for every proper subproduct G 0 , the projection map proj G 0 : G → G 0 has the properties that the image of Γ is dense and the map is faithful on Γ.
a lattice in a product of locally compact second countable groups.
Let κ = κ 1 × • • • × κ k be an admissible probability measure on G.When G is a semisimple group, we take κ j to be K j -invariant where K j is a maximal compact subgroup if G j is connected and Let µ be the admissible probability measure on Γ such that the Γ action on the (G, κ)-Poisson boundary is µ-stationary [Mar91].
A stationary action of Γ is an action which is µ-stationary for some µ coming from a product measure κ on G.

Stationary Actions of Commensurators
Definition 1.3.Let Γ < G be a lattice in a locally compact second countable group.A countable group Λ < G is a commensurator of Γ when Γ ⊆ Λ and for all λ ∈ Λ the group Γ ∩ λΓλ −1 has finite index in both Γ and λΓλ −1 .
We first establish that dense commensurators are, in a suitable sense, lattices in their own right: Theorem 1.4.Let Γ < G be a strongly irreducible lattice in a semisimple group G with trivial center of higher-rank (meaning G is either a simple higher-rank connected Lie group or G is a product of at least two simple factors).
Let Λ be a dense commensurator of Γ.
Then there exists a locally compact totally disconnected nondiscrete group H such that Λ can be embedded as a strongly irreducible lattice in G × H.

Proof.
The group H will be the relative profinite completion Λ/ /Γ (see e.g.[CP17] Section 6 for details, we indicate here the facts needed for the proof).Define the map τ : Λ → Symm(Λ/Γ) sending λ to its action on Λ/Γ as a symmetry of Λ/Γ.Theorem 6.3 in [CP17] states that H = τ (Λ) is a locally compact totally disconnected group and K = τ (Γ) is a compact open subgroup.
Let N = ker(τ ) be the kernel of τ .Then N ∩ Γ ⊳ Γ and N ⊳ Λ.By Margulis' Normal Subgroup Theorem, N ∩ Γ is either finite or has finite index in Γ.Since G is center-free, if N is finite it is trivial.Suppose it has finite index.As N ⊳ Λ then N ⊳ Λ = G.The only normal subgroups of G are finite or are proper subproducts.Since N contains a finite index subgroup of Γ, it is infinite, but then N must be contained in a proper subproduct so a finite index subgroup of Γ would be contained in a proper subproduct, contradicting that Γ is strongly irreducible.So the kernel of τ is trivial.Proposition 6.1.2[CP17] states that H will be discrete if and only if Λ normalizes a finite index subgroup of Γ.If Γ 0 is finite index in Γ and Γ 0 ⊳ Λ then Γ 0 ⊳ G as Γ 0 is discrete and Λ is dense, but this is impossible.So H is nondiscrete.
Clearly we can embed Λ → G × H diagonally and faithfully (as the kernel of τ is trivial).Likewise we can embed Γ diagonally, and we can identify both with their images.
Let U be an open neighborhood of the identity in By construction, Λ projects densely to H.As Λ is dense in G by hypothesis, it projects densely to G from G × H.For any proper subproduct G 0 of G, Λ projects densely to G 0 since Γ does.Since Γ projects densely to G 0 × K, we have that Λ projects densely to G 0 × H. Thus Λ is strongly irreducible.This allows us to define: Definition 1.5.Let Γ < G be a lattice and Λ < G a dense commensurator.Let κ = κ G × κ H be an admissible probability measure on G × H. Then there is a probability measure µ on Λ so that the Λ action on the (G × H, κ)-Poisson boundary is µ-stationary.
An action of Λ is stationary when it is stationary for some µ coming from such a κ (i.e. when it is a stationary action if we treat Λ as a lattice).
We also need the converse of the above theorem; lattices can be treated as commensurators: Proposition 1.6.Let Λ < G × H be a strongly irreducible lattice, G and H locally compact second countable groups with H totally disconnected and K < H a compact open subgroup.
Then Γ is a strongly irreducible lattice in G and Γ (X, ν) is a stationary action.
Let κ = κ G × κ H be the admissible measure on G × H and µ Λ the probability measure on Λ such that the (G × H, κ)-Poisson boundary is µ Λ -stationary under the Λ-action and Λ Define µ Γ to be the probability measure on Γ so that (B G , β G ) is µ Γ -stationary under the Γ-action.
Since the boundary map B is Λ-equivariant, it is Γ-equivariant and so we conclude that meaning the action is stationary for Γ.

Stationary Factor Theorems
The intermediate factor theorems of Stuck-Zimmer [SZ94] and Bader-Shalom [BS06], as well as those of the author [CP17], [Cre17], all make assertions about the structure of an intermediate factor A between a measure-preserving G-space (X, ν) and the product of it with the Poisson boundary (B × X, β × ν).We now establish such a factor theorem for stationary actions.

The Stationary Joining
For a stationary G-space (X, ν), the product space (B × X, β × ν) will not in general be stationary (and indeed will only be when X is measure-preserving).Our factor theorem employs the stationary joining of Furstenberg-Glasner [FG10] in its place.The reader is referred to Glasner [Gla03] for a detailed exposition on joinings.

The Invariant Products Functor
To formulate the intermediate factor theorem for general stationary actions, we recall the invariant products functor, introduced in [BS06]: a product of locally compact second countable groups.For each j write Ǧj = i =j G i for the subproduct excluding G j .
For a G-space (Y, η) write (Y j , η j ) for the Ǧj -ergodic components.When G is the invariant products functor.

The Stationary Intermediate Factor Theorem for Product Groups
Theorem 2.3.

product of at least two locally compact second countable groups and µ
Let (X, ν) be an ergodic (G, µ)-space.Let (B, β) be the Poisson boundary for (G, µ).
Let (A, α) be an intermediate factor: where the maps are G-maps that compose to the natural projection map.
Then A is isomorphic to the relative independent joining of Proof.The invariants product functor (see [Cre17] Section 2.12) mapping a G-space to the product of the spaces of G j -ergodic components is relatively measure-preserving for stationary actions ([Cre17] Proposition 2.12.1; see also [BS06] Proposition 1.10).
The Gj ergodic components of B X are B j X j (the reader may verify this straightforward fact).Thus we have the commuting diagram of G-maps Since B is a contractive space, the map B X → X is relatively contractive hence so is the map A → X.As the downward maps are relatively measure-preserving, Theorem 2.41 in [Cre17] implies that A is isomorphic to the relative independent joining of Let (X, ν) be an ergodic (G, µ)-space such that proj j stab(x) is dense in G j for all j almost everywhere.Then (X, ν) is measure-preserving and weakly amenable.
Proof.The map s j : X → S(G j ) by x → proj G j stab(x) sending each x to a closed subgroup of G j gives rise to a random subgroup of G j ; realizing that stationary random subgroup, via Theorem 3.3 [Cre17], as the stabilizers of an action As Z is ergodic (since X is), this means Z is the trivial space.Theorem 4.12 in [Cre17] says that X → Z is a relatively measure-preserving extension (since X is stationary and Z = Z 1 × • • • × Z k is the product random subgroups functor of [Cre17] applied to X) hence we conclude that (X, ν) is in fact measure-preserving.Now consider the factor theorem when A is an affine space over (X, ν) following the approach pioneered by Stuck and Zimmer [SZ94] and used in [BS06], [CP17], [Cre17], etc.The reader is referred to [Cre17] Section 2 for details on affine spaces and their relation to strong and weak amenability of actions.
For any affine space A over (X, ν), as the action of G on the Poisson boundary is strongly amenable ([Zim84] Section 4.3), there are G-maps B X → A → X which compose to the projection map.In particular, the pushforward of the stationary measure on B × X endows A with the structure of a (G, µ)-space.
We consider the case when A is orbital over X (has the same stabilizer subgroups).Since the projection of the stabilizers are dense, the stabilizers of X j are G j almost surely meaning that each X j is trivial.Therefore A is isomorphic to the independent joining of X and Let a ∈ A. Then stab(a) = stab(x) for some x mapped to from a so almost every a has stabilizer that projects densely to each G j .As each A j is a G j -space, this means that the stabilizers for A j are G j almost everywhere.So the A j are all trivial meaning that A is isomorphic to X.
As this holds for all affine orbital A, it follows that X is weakly amenable.

Inducing Stationary Actions
Let Γ be an irreducible lattice in a locally compact second countable group G. Let F be a fundamental domain for G/Γ with associated cocycle α : G × F → Γ given by gf α(g, f ) inF .Write m for the Haar measure on G restricted to F and normalized to be a probability measure.

The Classical Induced Action
The classical induced action, due to Zimmer (e.g.[SZ94]), constructs a measure-preserving G-action from a measure-preserving Γ-action as follows: define G F × X by and then the measure m × ν is preserved by the G-action.
Much of the issue in finding a generalization to stationary actions lies in the fact that there is exactly one G-quasi-invariant measure on F × X which projects to ν on X and this measure is not stationary (unless ν is preserved by Γ) so our induced action will not have the property that its projection to X is ν.However, the projection will be in the same measure class.

The Stationary Induced Action
Let κ be an admissible probability measure on G and let (B, β) be the Poisson boundary for (G, κ).Write µ for the probability measure on Γ so that (B, β) is µ-stationary.Since the action of Γ on (B, β) is strongly amenable (as the action of G is and Γ is closed in G) there exists the boundary Γ-map π : B → P (X) (Zimmer [Zim84]; see e.g.[BS06] Theorem 2.14).
Let (X, ν) be a (Γ, µ)-space.Define the pointwise action G F × X as in the classical case.Define the probability measure ρ on F × X by Definition 3.1.The (G, κ)-space just defined is the induced stationary action from the (Γ, µ)space (X, ν).We will write G × Γ X for this space.

Properties of the Induced Action
Proposition 3.2.The measure on the induced stationary space projects to F as m and projects to X as a measure in the same class as ν (when ν is measure-preserving, it projects to ν itself ).
Proof.Observe that proj where m is the symmetric opposite of m.So proj X ρ is in the same measure class as ν since m * β is in the same measure class as β.
That proj F ρ = m is immediate.
Proposition 3.3.The induced stationary action is weakly amenable if and only if the action on (X, ν) is weakly amenable.
Proof.This is Proposition 2.6.1 in [Cre17].While that proposition is stated as holding for measurepreserving actions, the result it relies on, stated as Theorem 2.25 [Cre17], is actually a result of Zimmer [Zim77] which holds for quasi-invariant actions.
Proposition 3.4.If the induced stationary action is measure-preserving so is (X, ν).

The Stationary Intermediate Factor Theorem for Lattices
Corollary 3.5.
a strongly irreducible lattice in a product of at least two locally compact second countable groups.
Let Γ (X, ν) be an ergodic stationary action such that proj j stab(x) is dense in G j for all j almost everywhere.Then (X, ν) is measure-preserving and weakly amenable.
Proof.Let (Y, η) be the induced stationary action.Since stab(f, x) = f stab(x)f −1 , we have that for almost every y and all j, proj j stab(y) = G j .Then Y is measure-preserving by Corollary 2.4 hence X is by Proposition 3.4.

Projecting Actions to the Ambient Group
Theorem 4.1.Let Γ < G be a lattice in a locally compact second countable group and Λ < G a dense commensurator of Γ.
Let s : X → S(G) be a Λ-equivariant map.
Proof.The measure-preserving case, used in [CP17], is an easy consequence of density.The stationary case requires more care.Treat (X, ν) as a Γ-space and let π : As Λ is dense and G P (S(G)) continuously, then s * * π * β is in fact G-stationary as for every g ∈ G there exists λ n ∈ Λ with λ n → g and so As the barycenter map is G-equivariant, then bar s * * π * β is G-stationary and in particular there is a quasi-invariant action of G on (S(G), bar s * * π * β).Theorem 3.3 [Cre17] then gives an action G (Y, η) such that stab * η = bar s * * π * β = s * ν which may be taken to be ergodic since Λ (X, ν) is ergodic.
Corollary 4.2.Let Γ < G be a lattice in a locally compact second countable group and Λ < G a dense commensurator of Γ.Every stationary random subgroup of Λ becomes a stationary random subgroup of G under the closure map.
Theorem 4.3.Let G = j G j be a product of at least two simple locally compact second countable groups and let Γ < G be a strongly irreducible lattice.
Let s : X → S(G 0 ) be a Γ-equivariant map to the space of closed subgroups of some proper subproduct G 0 ⊳ G.
Then there exists an ergodic G 0 -space (Y, η) such that stab * η = s * ν.There exists Proof.Since Γ is irreducible, H acts ergodically on ((G × H)/Γ, m) where m is the unique invariant probability measure.Let K ⊆ H be compact with nonempty interior.Then m((U ×K)Γ) > 0. The random ergodic theorem [Kak51] tells us for m-almost every z ∈ (G × H)/Γ and κ N -almost every Γ infinitely often and β (ωn) (kE) = 1.So there exists {n j } so that ω −1 Write proj 0 : Γ → G 0 for the (faithful) projection map.If γ n ∈ Γ such that proj 0 γ n → e in G 0 then s(γ n x) → s(x) as G 0 acts continuously on S(G 0 ).Therefore Let E be a positive measure subset of Z. Lemma 4.4 gives γ n ∈ Γ such that proj 0 γ n → e in G 0 and β(γ n p−1 (E)) → 1 (taking U n to be a decreasing sequence of open neighborhoods of e and ǫ to be 1

This means that if proj
is in the same class as s * * π * β.As Γ projects densely to G 0 , this means g 0 s * * π * β is in the same measure class as s * * π * β for every g 0 ∈ G 0 , i.e. that s * * π * β, and therefore also s * ν, is G 0 -quasi-invariant (and in fact stationary) so the result follows using Theorem 3.3 [Cre17].

Stabilizers of Lattices in Connected Groups
Let Γ (X, ν) be an ergodic stationary action.
Then either stab Γ (x) projects densely to each simple G k almost everywhere or stab Γ (x) is finite almost everywhere.
The proof will occupy this section.We begin with an observation of Vershik [Ver11]: Proposition 5.2.Let Λ be a countable group and Λ (X, ν) a quasi-invariant action.

Howe-Moore Groups
Recall that a group is Howe-Moore when for every unitary representation without invariant vectors, the matrix coefficients vanish at infinity [HM79].
Lemma 5.3.Let G be a locally compact second countable group with the Howe-Moore property and G (Y, η) a quasi-invariant action on a probability space.
If there exists a subgroup Proof.First, observe that for any f ∈ L 2 (Y, η) and q ∈ Q, and therefore dqη dη (y) = 1 for y ∈ Fix Q and q ∈ Q.
Let π be the unitary representation of G on L 2 (Y, η): π(g)f (y) = f (g −1 y) dgη dη (y).Let P be the orthogonal projection from L 2 (Y, η) to the closed G-invariant subspace I of π-invariant vectors.Then π restricted to L 2 (Y, η) ⊖ I has no invariant vectors.
Suppose E is not a G-invariant set.Then there exists g ∈ G with η(gE \ E) > 0. Writing 1 E for the characteristic function Therefore for y ∈ gE \ E we have 0 = dgη dη (y) contradicting that the G action is quasi-invariant.Therefore E is G-invariant as a set.
Since every subset of Fix Q is a G-invariant set, G fixes almost every point in it.

Connected Lie Groups
Lemma 5.4.Let G be a simple connected Lie group and G (Z, ζ) be a quasi-invariant (not necessarily ergodic) action.Let C(z) be the connected component of stab(z).Let d ≥ 0 be a nonnegative integer.
If dim C(z) ≤ d almost everywhere then one of the following holds: Fubini's theorem tells us that for some x there is a positive measure set of y so that ζ({x : Suppose η({y ∈ Y : stab(y) is not compact and not discrete}) > 0. Apply Lemma 5.4 to conclude that either η({y ∈ Y : stab(y) = G}) > 0 or that G (Y 2 , η 2 ) has the property that for η × ηalmost every y 2 ∈ Y 2 it holds that stab(y 2 ) is at least one of compact, discrete, all of G, or has dimension ≤ d − 1. Repeating this process by applying Lemma 5.4 to (Y 2 , η 2 ), since d is finite, we conclude that there is some finite number t of applications so that for m = 2 t we have that the diagonal action G (Y m , η m ) has the property that almost every stabilizer is at least one of compact, discrete or all of G.
By ergodicity then this holds almost everywhere but that contradicts that (Y, η) is nontrivial.
Proposition 5.6.Let G be a semisimple connected Lie group with trivial center and Λ < G a countable subgroup and G (Y, η) a quasi-invariant action.
If the G-stabilizers are discrete almost everywhere then Λ (Y, η) is essentially free.
Proof.Suppose the Λ-action is not essentially free.Then there exists λ ∈ Λ, λ = e, with ν(Fix λ) > 0. Since G is connected and λ / ∈ Z(G), the centralizer subgroup of λ does not contain an open neighborhood the identity in G.So there exists g n → e in G such that g n λg −1 n = λ for all n.Note that, writing E = Fix λ, we have η(g n E △E) → 0. Take a subsequence along which n we have that λy = y and g n λg −1 n y = y for all n, hence g n λg −1 n ∈ stab G (y) and λ ∈ stab G (y).But g n λg −1 n = λ and g n λg −1 n → λ contradicting that stab G (y) is discrete almost everywhere.

Miscellany
Lemma 5.7.Let G be a locally compact second countable group and K a compact group.Let Lemma 5.8.Let H be a group and L < H a subgroup.If τ is a random subgroup of H that is supported on subgroups of L then τ is supported on subgroups of N for some N ⊳ H with N ⊆ L.
Proof.Since τ is supported on S(L), for τ -almost every Q ∈ S(H) in fact Q ⊆ L. As τ is H-quasiinvariant, for all h and τ -almost every Q we have hQh −1 ⊆ L as well.Then Q ⊆ ∩ h h −1 Lh for almost every Q.Set N = ∩ h h −1 Lh which is normal in H and contained in L. Then Q ⊆ N almost everywhere as claimed.
Recall that a group is locally finite when every finitely generated subgroup of it is finite.Proposition 5.9.Let Γ be a virtually torsion-free countable discrete group and Γ (X, ν) a quasi-invariant action with locally finite stabilizers almost everywhere.Then the stabilizers are finite almost everywhere.
Proof.Since Γ is virtually torsion-free, there is a finite upper bound ℓ for the length of every strictly increasing chain of finite subgroups.
As stab(x) is locally finite, if it is infinite then it contains an infinite abelian subgroup A (Hall and Kulatilaka [HK64]).But A must be of the form ⊕ ∞ n=1 F n for some nontrivial finite subgroups F n and then A N = ⊕ N n=1 F n would be a strictly increasing chain of finite subgroups of infinite length.Therefore stab(x) must in fact be finite.

Lattices and Projected Actions
Proposition 5.10.
Let Γ < G be a strongly irreducible lattice.
Then either stab Γ (x) is locally finite almost everywhere or stab Γ (x) has dense projections to each simple G j almost everywhere.
Proof.For each j ∈ {1, 2, . . ., k}, the projection map proj G j : G → G j has the property that the image of Γ is dense in G j and the map is faithful on Γ. Define the map s j : X → S(G j ) by s j (x) = proj G j stab Γ (x).
Define the sets Note that for j ∈ J d we have that stab * η j is a point mass δ Q for some Q ∈ S(G j ).As stab * η j is Γ-invariant this means γQγ −1 = Q for all γ.As Γ is dense ) is essentially free since the projection map is faithful, in which case the proof is complete.So we proceed with the premise that proj G j stab(x) is dense in G j almost everywhere for j ∈ J d .
From here on, we assume that J c = ∅ since if it is empty then the proof is complete as the projections of the stabilizers are dense to each G j almost everywhere.

Define the groups
so that G = G c × G d (after appropriate rearrangement).
For j ∈ J c , Proposition 5.5 says there exists a positive integer m j so that for the diagonal action G j (Y m j j , η m j j ) has the property that almost every stabilizer is compact or is discrete.Set m = max(m j : j ∈ J c ).Then almost every stabilizer of (Y j , η j ) m is discrete or compact.Hence, for almost every (y 1 , . . ., y m ) ∈ Y m j we have that ∩ m i=1 stab(y i ) is discrete or compact.Then ∩ m i=1 proj G j stab Γ (x i ) is discrete or compact for ν m -almost every (x 1 , . . ., x m ) ∈ X m .So for ν m -almost every (x 1 , . . ., x m ) we have proj G j ∩ m i=1 stab Γ (x i ) is discrete or compact.

Discrete Stabilizers
The set D = {z ∈ Y m j : stab(z) is discrete} is G j -invariant, and if it is positive measure we may consider G j (D, (η m ) * ) where (η m ) * is η m restricted to D and normalized to be a probability measure.Applying Proposition 5.6, we determine that stab(z) ∩ proj G j Γ = {e} for a.e.z ∈ D. Now if proj G j stab Γ (x 1 , . . ., x m ) ∩ proj G j Γ = {e} then proj G j stab Γ (x 1 , . . ., x m ) = {e} and, as the projection is faithful on Γ, then stab Γ (x 1 , . . ., x m ) = {e}.

Compact Stabilizers
For z / ∈ D, the corresponding proj G j stab Γ (x 1 , . . ., x m ) is contained in a compact subgroup.Therefore proj G j stab Γ (x 1 , . . ., x m ) is contained in a compact subgroup of G j a.e. for all j ∈ J c .

The Diagonal Action Γ (X, ν) m
Every stationary action of Γ either has finite stabilizers or finite index stabilizers.
Proof.Write G = G 0 × H where G 0 is connected and H is totally disconnected.If H is trivial then Theorem 6.2 gives the conclusion; if not then by Proposition 1.6 the projection of Γ to G is a dense commensurator of a strongly irreducible lattice Γ 0 < G 0 and any stationary action of Γ is stationary for Γ 0 so Corollary 6.4 then gives the result (the case when G is a simple connected higher-rank group is covered by [BH19]).

A. The Stationary Intermediate Factor Theorem for Dense Commensurators
For completeness, we include a stationary form of the factor theorem for commensurators, though as we did not need it above we include it as an appendix.
Such a theorem will be needed to handle e.g.actions of tree lattices, though we do not elaborate nor attempt that here.
Theorem A.2. Let Γ < G be a lattice in a locally compact second countable group and Λ < G a dense commensurator.
Let Λ (X, ν) be an ergodic Λ-space such that the restriction of the action to Γ is stationary.
Let (B, β) be the Poisson boundary of G. Let (Z, ζ) be a (Γ, µ)-space with Γ-maps that compose to the projection map.
Proof.We remark that what follows is near identical to the proof of Theorem 4.19 in [CP17] but as the results there are stated for measure-preserving (X, ν), we cannot cite them directly.
on that positive measure set and we are in the first case.Proceed now assuming we are not in the first case.Let M = {z ∈ Z : stab(z) is not compact and not discrete } For almost every x, y ∈ M we have C(x) = C(y) by the above and that dim C(x) > 0 and dim C(y) > 0 since they are nondiscrete.Then dim C(x) ∩ C(y) < dim C(x) ≤ d.Proposition 5.5.Let G be a connected simple Lie group.Let G (Y, η) be an ergodic nontrivial G-space.Then there exists a positive integer m such that the diagonal action G (Y m , η m ) has the property that for almost every z ∈ Y m the stabilizer subgroup stab(z) is discrete or compact.Proof.Write C(y) for the connected component of stab(y).Set d = sup dim C(y) ≤ dim G.