Invariant random subgroups of linear groups

Invariant Random Subgroups of Inductive Limits of Finite Alternating Groups

We classify the ergodic invariant random subgroups of inductive limits of finite alternating groups.

Unimodularity of Invariant Random Subgroups

An invariant random subgroup H ≤ G is a random closed subgroup whose law is invariant to conjugation by all elements of G. When G is locally compact and second countable, we show that for every invariant random subgroup H ≤ G there almost surely exists an invariant measure on G/H. Equivalently, the modular function of H is almost surely equal to the modular function of G, restricted to H. We us...

Invariant random subgroups of semidirect products

We study invariant random subgroups (IRSs) of semidirect products G = AoΓ. In particular, we characterize all IRSs of parabolic subgroups of SLd(R), and show that all ergodic IRSs of Rdo SLd(R) are either of the form RdoK for some IRS of SLd(R), or are induced from IRSs of Λo SL(Λ), where Λ < Rd is a lattice.

Kesten’s theorem for Invariant Random Subgroups

An invariant random subgroup of the countable group Γ is a random subgroup of Γ whose distribution is invariant under conjugation by all elements of Γ. We prove that for a nonamenable invariant random subgroup H , the spectral radius of every finitely supported random walk on Γ is strictly less than the spectral radius of the corresponding random walk on Γ/H . This generalizes a result of Keste...

Random Normal Subgroups of Free Profinite Groups