# Journal of the European Mathematical Society

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**Volume 18, Issue 12, 2016, pp. 2733–2784**

**DOI: 10.4171/JEMS/652**

Published online: 2016-11-21

Orbit equivalence and Borel reducibility rigidity for profinite actions with spectral gap

Adrian Ioana^{[1]}(1) University of California, San Diego, United States

We study equivalence relations $\mathcal R(\Gamma\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $\Gamma\curvearrowright G$ is free and has spectral gap, we describe precisely when $\mathcal R(\Gamma\curvearrowright G)$ is orbit equivalent or Borel reducible to another such equivalence relation $\mathcal R(\Lambda\curvearrowright H)$. As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of $SL_2(\mathbb Z)$ and its non-amenable subgroups (e.g. $\mathbb F_n$, with $2\leqslant n\leqslant\infty$) whose orbit equivalence relations are mutually not orbit equivalent and not Borel reducible. In particular, we show that if $S$ and $T$ are distinct sets of primes, then the orbit equivalence relations associated to the actions $SL_2(\mathbb Z)\curvearrowright\prod_{p\in S}SL_2(\mathbb Z_p)$ and $SL_2(\mathbb Z)\curvearrowright\prod_{p\in T}SL_2(\mathbb Z_p)$ are neither orbit equivalent nor Borel reducible. This settles a conjecture of S. Thomas [Th01,Th06]. Other applications include the first calculations of outer automorphism groups for concrete treeable p.m.p. equivalence relations, and the first concrete examples of free ergodic p.m.p. actions of $\mathbb F_{\infty}$ whose orbit equivalence relations have trivial fundamental group.

*Keywords: *Spectral gap, rigidity, orbit equivalence, Borel reducibility, equivalence relations, profinite actions, outer automorphism group, II$_1$ factor

Ioana Adrian: Orbit equivalence and Borel reducibility rigidity for profinite actions with spectral gap. *J. Eur. Math. Soc.* 18 (2016), 2733-2784. doi: 10.4171/JEMS/652