Invariant random subgroups of linear groups

Abstract

Let $\Gamma < \mathrm{GL}_n(F)$ be a countable non-amenable linear group with a simple, center free Zariski closure, $\mathrm{Sub}(\Gamma)$ the space of all subgroups of $\Gamma$ with the, compact, metric, Chabauty topology. An invariant random subgroup (IRS) of $\Gamma$ is a conjugation invariant Borel probability measure on $\mathrm{Sub}(\Gamma)$. An $\mathrm{IRS}$ is called nontrivial if it does not have an atom in the trivial group, i.e. if it is nontrivial almost surely. We denote by $\mathrm{IRS}^{0}(\Gamma)$ the collection of all nontrivial $\mathrm{IRS}$ on $\Gamma$. We show that there exits a free subgroup $F < \Gamma$ and a non-discrete group topology $\mathrm{St}$ on $\Gamma$ such that for every $\mu \in \mathrm{IRS}^{0}(\Gamma)$ the following properties hold: (i) $\mu$-almost every subgroup of $\Gamma$ is open. (ii) $F \cdot \Delta = \Gamma$ for $\mu$-almost every $\Delta \in \mathrm{Sub}(\Gamma)$. (iii) $F \cap \Delta$ is infinitely generated, for every open subgroup. (iv) The map $\Phi: (\mathrm{Sub}(\Gamma),\mu) \rightarrow (\mathrm{Sub}(F),\Phi_* \mu)$ given by $\Delta \mapsto \Delta \cap F$, is an $F$-invariant isomorphism of probability spaces. We say that an action of $\Gamma$ on a probability space, by measure preserving transformations, is almost surely non free (ASNF) if almost all point stabilizers are non-trivial. As a corollary of the above theorem we show that the product of finitely many ANSF $\Gamma$-spaces, with the diagonal $\Gamma$ action, is ASNF. Let $\Gamma < \mathrm{GL}_n(F)$ be a countable linear group, $A \lhd \Gamma$ the maximal normal amenable subgroup of $\Gamma$. We show that if $\mu \in \mathrm{IRS}(\Gamma)$ is supported on amenable subgroups of $\Gamma$ then in fact it is supported on $\mathrm{Sub}(A)$. In particular if $A(\Gamma) = \langle e \rangle$ then $\Delta = \langle e \rangle, \mu$ almost surely.