Invariant random subgroups of linear groups Academic Article uri icon

abstract

  • An "invariant random subgroup" (IRS) of a countable discrete group $\Gamma$ is, by definition, a conjugation invariant probability measure on the compact metric space $Sub(\Gamma$) of all subgroups of $\Gamma$. We denote by $IRS(\Gamma)$ the collection of all such invariant measures. Theorem: Let $\Gamma < GL_n(F)$ be a countable non-amenable linear group with a simple, center free Zariski closure. There exits a non-discrete group topology ${\mathcal{M}}$e on $\Gamma$ such that for every $\mu \in IRS(\Gamma)$, $\mu$-almost every subgroup $\langle e \rangle \ne \Delta \in Sub(\Gamma)$ is open. Moreover there exits a free subgroup $F < \Gamma$ with the following properties: (i) $F \cap \Delta$ is an infinitely generated free group, for every open subgroup $\Delta \in Sub(\Gamma)$. (ii) $F \cdot \Delta = \Gamma$, for every $\mu \in IRS(\Gamma)$ and $\mu$ a.e. $\langle e \rangle \ne \Delta \in Sub(\Gamma).$ (iii) The map $\Phi: (Sub(\Gamma),\mu) \rightarrow (Sub(F),\Phi_* \mu)$ given by $\Delta \mapsto \Delta \cap F$ is an $F$-invariant isomorphism of probability spaces, for every $\mu \in IRS(\Gamma)$. 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. For probability measure preserving actions of $\Gamma$ as in the main theorem we prove the following: Corollary 1: The product of two ASNF $\Gamma$ spaces with the diagonal $\Gamma$ action is still ASNF. Corollary 2: If the stabilizers of almost all points are amenable then the action is essentially free.

publication date

  • January 1, 2017