Generic IRS in free groups, after Bowen
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- by Amichai Eisenmann and Yair Glasner PDF
- Proc. Amer. Math. Soc. 144 (2016), 4231-4246 Request permission
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.
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Additional Information
- Amichai Eisenmann
- Affiliation: Department of Mathematics. Ben-Gurion University of the Negev. P.O.B. 653, Be’er Sheva 84105, Israel
- MR Author ID: 831158
- Email: amichaie@math.bgu.ac.il
- Yair Glasner
- Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er Sheva 84105, Israel
- MR Author ID: 673281
- ORCID: 0000-0002-6231-3817
- Email: yairgl@math.bgu.ac.il
- Received by editor(s): July 4, 2014
- Received by editor(s) in revised form: May 27, 2015, and September 21, 2015
- Published electronically: June 10, 2016
- Communicated by: Nimish Shah
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4231-4246
- MSC (2010): Primary 37A20; Secondary 20B22, 37A15, 43A07
- DOI: https://doi.org/10.1090/proc/13020
- MathSciNet review: 3531175