Generic IRS in free groups, after Bowen

Abstract

Let $E$ be a measure preserving equivalence relation, with countable equivalence classes, on a standard Borel probability space $(X,B,\mu)$. Let $([E],d_{u})$ be the the (Polish) full group endowed with the uniform metric. If $F_r = \langle s_1, \ldots, s_r \rangle$ is a free group on $r$-generators and $\alpha \in \operatorname{Hom}(F_r,[E])$ then the stabilizer of a $\mu$-random point $\alpha(F_r)_x$ is a random subgroup of $F_r$ whose distribution is conjugation invariant. Such an object is known as an "invariant random subgroup" or an IRS for short. Bowen's generic model for IRS in $F_r$ is obtained by taking $\alpha$ to be a Baire generic element in the Polish space $\operatorname{Hom}(F_r, [E])$. The "lean aperiodic model" is a similar model where one forces $\alpha(F_r)$ to have infinite orbits by imposing that $\alpha(s_1)$ be aperiodic. In this setting we show that for $r < \infty$ the generic IRS $\alpha(F_r)_x$ is of finite index in $F_r$ a.s. if and only if $E = E_0$ is the hyperfinite equivalence relation. For any ergodic equivalence relation we show that a generic IRS coming from the lean aperiodic model is co-amenable and core free. Finally, we consider the situation where $\alpha(F_r)$ is highly transitive on almost every orbit and in particular the corresponding IRS is supported on maximal subgroups. Using a result of Le-Ma\^{i}tre we show that such examples exist for any aperiodic ergodic $E$ of finite cost. For the hyperfinite equivalence relation $E_0$ we show that high transitivity is generic in the lean aperiodic model.