Invariant random subgroups of groups acting on hyperbolic spaces

Abstract

Suppose that a group G acts non-elementarily on a hyperbolic space S and does not fix any point of ∂S. A subgroup H ≤ G is geometrically dense in G if the limit sets of H and G on ∂S coincide and H does not fix any point of ∂S. We prove that every invariant random subgroup of G is either geometrically dense or contained in the elliptic radical (i.e., the maximal normal elliptic subgroup of G). In particular, every ergodic measure preserving action of an acylindrically hyperbolic group on a Borel probability space (X, µ) either has finite stabilizers µ-almost surely or otherwise the stabilizers are very large (in particular, acylindrically hyperbolic) µ-almost surely.

Stationary random subgroups in negative curvature

“…Lattices in the group of isometries Is(X) as well as normal subgroups of such lattices have full limit sets. The same is true for invariant random subgroups of the group Is(X) as was shown in [ABB + 20, Proposition 11.3] and [Osi17]. It turns out that this fact extends to discrete stationary random subgroups.…”

Local weak$^{*}$-Convergence, algebraic actions, and a max-min principle

“…Related results had also been proved before Abért-Glasner-Virag by Aldous-Lyons [5], Bergeron-Gaboriau [11], and Vershik [57]. The study of invariant random subgroups has been quite active in recent years with connections to L 2 -invariants [1], geometric group theory [9,48],…”

Betti numbers of Shimura curves and arithmetic three-orbifolds

“…For the proof we use the estimates developped in [10] to show that any weak-* limit of the sequence µ Γn is supported on elementary subgroups. By [23] the only IRS supported on this set is the trivial IRS, hence the theorem. We carry out the second step of this scheme of proof in detail in Proposition 2.4, which is valid for all sequences of lattices in proper Gromov-hyperbolic spaces.…”

“…We assume that the action of G on X is non-elementary. The elliptic radical of G can then be defined as its unique maximal normal compact subgroup (see [23,Proposition 3.4]; in our context, by properness of X means that bounded elements are the same as compact ones). The following lemma is a special case of [23,Theorem 1.5].…”