Invariant random subgroups of linear groups

Let $Γ< \mathrm{GL}_n(F)$ be a countable non-amenable linear group with a simple, center free Zariski closure, $\mathrm{Sub}(Γ)$ the space of all subgroups of $Γ$ with the, compact, metric, Chabauty topology. An invariant random subgroup (IRS) of $Γ$ is a conjugation invariant Borel probability measure on $\mathrm{Sub}(Γ)$. 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 $\mathrm{IRS}^{0}(Γ)$ the collection of all nontrivial $\mathrm{IRS}$ on $Γ$. We show that there exits a free subgroup $F < Γ$ and a non-discrete group topology $\mathrm{St}$ on $Γ$ such that for every $μ\in \mathrm{IRS}^{0}(Γ)$ the following properties hold: (i) $μ$-almost every subgroup of $Γ$ is open. (ii) $F \cdot Δ= Γ$ for $μ$-almost every $Δ\in \mathrm{Sub}(Γ)$. (iii) $F \cap Δ$ is infinitely generated, for every open subgroup. (iv) The map $Φ: (\mathrm{Sub}(Γ),μ) \rightarrow (\mathrm{Sub}(F),Φ_* μ)$ given by $Δ\mapsto Δ\cap F$, is an $F$-invariant isomorphism of probability spaces. We say that an action of $Γ$ 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 $Γ$-spaces, with the diagonal $Γ$ action, is ASNF. Let $Γ< \mathrm{GL}_n(F)$ be a countable linear group, $A \lhd Γ$ the maximal normal amenable subgroup of $Γ$. We show that if $μ\in \mathrm{IRS}(Γ)$ is supported on amenable subgroups of $Γ$ then in fact it is supported on $\mathrm{Sub}(A)$. In particular if $A(Γ) = \langle e \rangle$ then $Δ= \langle e \rangle, μ$ almost surely.

No reviews yet.

No comments yet.