# Generic IRS in free groups, after Bowen Academic Article

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### 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