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

Abstract

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.