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- W1533409598 abstract "We study equivalence relations $mathcal R(Gammacurvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $Gammacurvearrowright G$ is free and has spectral gap, we describe precisely when $mathcal R(Gammacurvearrowright G)$ is orbit equivalent or Borel reducible to another such equivalence relation $mathcal R(Lambdacurvearrowright H)$. 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 $2leqslant nleqslantinfty$) 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 $SL_2(mathbb Z)curvearrowrightprod_{pin S}SL_2(mathbb Z_p)$ and $SL_2(mathbb Z)curvearrowrightprod_{pin T}SL_2(mathbb Z_p)$ 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." @default.
- W1533409598 created "2016-06-24" @default.
- W1533409598 creator A5070665706 @default.
- W1533409598 date "2013-09-12" @default.
- W1533409598 modified "2023-09-27" @default.
- W1533409598 title "Orbit equivalence and Borel reducibility rigidity for profinite actions with spectral gap" @default.
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