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We study equivalence relations $\mathcal R(\Gamma\curvearrowright G)$ thatarise from left translation actions of countable groups on their profinitecompletions. Under the assumption that the action $\Gamma\curvearrowright G$ isfree and has spectral gap, we describe precisely when $\mathcalR(\Gamma\curvearrowright G)$ is orbit equivalent or Borel reducible to anothersuch equivalence relation $\mathcal R(\Lambda\curvearrowright H)$. As aconsequence, we provide explicit uncountable families of free ergodicprobability measure preserving (p.m.p.) profinite actions of $SL_2(\mathbb Z)$and its non-amenable subgroups (e.g. $\mathbb F_n$, with $2\leqslantn\leqslant\infty$) whose orbit equivalence relations are mutually not orbitequivalent and not Borel reducible. In particular, we show that if $S$ and $T$are distinct sets of primes, then the orbit equivalence relations associated tothe actions $SL_2(\mathbb Z)\curvearrowright\prod_{p\in S}SL_2(\mathbb Z_p)$and $SL_2(\mathbb Z)\curvearrowright\prod_{p\in T}SL_2(\mathbb Z_p)$ areneither orbit equivalent nor Borel reducible. This settles a conjecture of S.Thomas \cite{Th06}. Other applications include the first calculations of outerautomorphism groups for concrete treeable p.m.p. equivalence relations, and thefirst concrete examples of free ergodic p.m.p. actions of $\mathbb F_{\infty}$whose orbit equivalence relations have trivial fundamental group.

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