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

We study equivalence relations $\mathcal{R}(\mathrm{\Gamma }\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $\mathrm{\Gamma }\curvearrowright G$ 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 $\mathcal{R}(\mathrm{\Lambda }\curvearrowright H)$. As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of $S{L}_2(\mathbb{Z})$ 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 $S{L}_2(\mathbb{Z})\curvearrowright {\prod }_{p\in S}S{L}_2({\mathbb{Z}}_p)$ and $S{L}_2(\mathbb{Z})\curvearrowright {\prod }_{p\in T}S{L}_2({\mathbb{Z}}_p)$ are neither orbit equivalent nor Borel reducible. This settles a conjecture of S. Thomas \cite{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 ${\mathbb{F}}_{\mathrm{\infty }}$ whose orbit equivalence relations have trivial fundamental group.