Generic IRS in free groups, after Bowen Academic Article uri icon

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\leftslice\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) …

publication date

  • January 1, 2016