Generic IRS in free groups, after Bowen

Let E be a measure preserving equivalence relation, with countable equivalence classes, on a standard Borel probability space (X,B, μ). Let ([E], d_u) be the (Polish) full group endowed with the uniform metric. If 𝔽_r = 〈s₁, …, s_r〉 is a free group on r-generators and α ∈ Hom(𝔽_r, [E]), then the stabilizer of a μ-random point α(𝔽_r)_x ⊲ 𝔽_r 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 𝔽r is obtained by taking α to be a Baire generic element in the Polish space Hom(𝔽r, [E]). The lean aperiodic model is a similar model where one forces α(𝔽_r) to have infinite orbits by imposing that α(s₁) be aperiodic. In Bowen’s setting we show that for r < ∞ the generic IRS α(𝔽_r)_x⊲𝔽_r is of finite index almost surely if and only if E = E₀ 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) 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 E0 we show that high transitivity is generic in the lean aperiodic model.