  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 \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 $\mathbb F_{\infty}$
whose orbit equivalence relations have trivial fundamental group.
