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 $(X,\mathcal {B},\mu )$. Let $([E],d_{u})$ be the (Polish) full group endowed with the uniform metric. If $\mathbb {F}_r = \langle s_1, \ldots , s_r \rangle$ is a free group on $r$-generators and $\alpha \in \mathrm {Hom}(\mathbb {F}_r,[E])$, then the stabilizer of a $\mu$-random point $\alpha (\mathbb {F}_r)_x \ltcc \mathbb {F}_r$ is a random subgroup of $\mathbb {F}_r$ 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 $\mathbb {F}_r$ is obtained by taking $\alpha$ to be a Baire generic element in the Polish space ${\mathrm {Hom}}(\mathbb {F}_r, [E])$. The lean aperiodic model is a similar model where one forces $\alpha (\mathbb {F}_r)$ to have infinite orbits by imposing that $\alpha (s_1)$ be aperiodic.

In Bowen’s setting we show that for $r < \infty$ the generic IRS $\alpha (\mathbb {F}_r)_x \ltcc \mathbb {F}_r$ is of finite index almost surely 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 $\alpha (\mathbb {F}_r)$ is highly transitive on almost every orbit and in particular the corresponding IRS is supported on maximal subgroups. Using a result of Le Maître 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.