Generic IRS in free groups, after Bowen

Abstract

Let E E be a measure preserving equivalence relation, with countable equivalence classes, on a standard Borel probability space ( X , B , μ ) (X,\mathcal {B},\mu ) . Let ( [ E ] , d u ) ([E],d_{u}) be the (Polish) full group endowed with the uniform metric. If F r = ⟨ s 1 , … , s r ⟩ \mathbb {F}_r = \langle s_1, \ldots , s_r \rangle is a free group on r r -generators and α ∈ H o m ( F r , [ E ] ) \alpha \in \mathrm {Hom}(\mathbb {F}_r,[E]) , then the stabilizer of a μ \mu -random point \alpha (\mathbb {F}_r)_x \ltcc \mathbb {F}_r is a random subgroup of F r \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 F r \mathbb {F}_r is obtained by taking α \alpha to be a Baire generic element in the Polish space H o m ( F r , [ E ] ) {\mathrm {Hom}}(\mathbb {F}_r, [E]) . The lean aperiodic model is a similar model where one forces α ( F r ) \alpha (\mathbb {F}_r) to have infinite orbits by imposing that α ( s 1 ) \alpha (s_1) be aperiodic. In Bowen’s setting we show that for r > ∞ 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 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 α ( F r ) \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 E of finite cost. For the hyperfinite equivalence relation E 0 E_0 we show that high transitivity is generic in the lean aperiodic model.