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Generic IRS in free groups, after Bowen


Authors: Amichai Eisenmann and Yair Glasner
Journal: Proc. Amer. Math. Soc. 144 (2016), 4231-4246
MSC (2010): Primary 37A20; Secondary 20B22, 37A15, 43A07
Published electronically: June 10, 2016
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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)$ 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 \leftslice \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
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
Email: yairgl@math.bgu.ac.il

DOI: https://doi.org/10.1090/proc/13020
Keywords: IRS, free groups
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
Article copyright: © Copyright 2016 American Mathematical Society