Invariant random subgroups of linear groups

Let $\mathrm{\Gamma }<{\mathrm{G}\mathrm{L}}_n(F)$ be a countable non-amenable linear group with a simple, center free Zariski closure, $\mathrm{S}\mathrm{u}\mathrm{b}(\mathrm{\Gamma })$ the space of all subgroups of $\mathrm{\Gamma }$ with the, compact, metric, Chabauty topology. An invariant random subgroup (IRS) of $\mathrm{\Gamma }$ is a conjugation invariant Borel probability measure on $\mathrm{S}\mathrm{u}\mathrm{b}(\mathrm{\Gamma })$. An $\mathrm{I}\mathrm{R}\mathrm{S}$ 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 ${\mathrm{I}\mathrm{R}\mathrm{S}}^0(\mathrm{\Gamma })$ the collection of all nontrivial $\mathrm{I}\mathrm{R}\mathrm{S}$ on $\mathrm{\Gamma }$. We show that there exits a free subgroup $F<\mathrm{\Gamma }$ and a non-discrete group topology $\mathrm{S}\mathrm{t}$ on $\mathrm{\Gamma }$ such that for every $\mu \in $\mathrm{I}\mathrm{R}\mathrm{S}$^{0}(\Gamma)$thefollowingpropertieshold:(i)$\mu$-almost every subgroup of $\mathrm{\Gamma }$ is open. (ii) $F\cdot \mathrm{\Delta }=\mathrm{\Gamma }$ for $\mu$-almost every $\mathrm{\Delta }\in \mathrm{S}\mathrm{u}\mathrm{b}(\mathrm{\Gamma })$. (iii) $F\cap \mathrm{\Delta }$ is infinitely generated, for every open subgroup. (iv) The map $\Phi: ($\mathrm{S}\mathrm{u}\mathrm{b}$(\Gamma),\mu) \rightarrow ($\mathrm{S}\mathrm{u}\mathrm{b}$(F),\Phi_* \mu)$givenby$\Delta \mapsto \Delta \cap F$,isan$F$-invariant isomorphism of probability spaces. We say that an action of $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$-spaces, with the diagonal $\mathrm{\Gamma }$ action, is ASNF. Let $\mathrm{\Gamma }<{\mathrm{G}\mathrm{L}}_n(F)$ be a countable linear group, $A⊲\mathrm{\Gamma }$ the maximal normal amenable subgroup of $\mathrm{\Gamma }$. We show that if $\mu \in $\mathrm{I}\mathrm{R}\mathrm{S}$(\Gamma)$issupportedonamenablesubgroupsof$\Gamma$ then in fact it is supported on $\mathrm{S}\mathrm{u}\mathrm{b}(A)$. In particular if $A(\Gamma) = \langle e \rangle$then$\Delta = \langle e \rangle, \mu$ almost surely.