2013/09/12 by Adrian Ioana, Ioana, Adrian
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #math.DS #math.GR #math.OA
paper · pdf · doi:10.48550/arxiv.1309.3026
v2: added a necessary condition in the statement of Theorem 4.4 and corrected the proofs of Theorems 4.4 and 7.1 accordingly; v3: improved exposition; v4: final version
arxiv created 2015/07/30 · arxiv updated 2015/08/03
We study equivalence relations \mathcal R(Γ\curvearrowright G) that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action Γ\curvearrowright G is free and has spectral gap, we describe precisely when \mathcal R(Γ\curvearrowright G) is orbit equivalent or Borel reducible to another such equivalence relation \mathcal R(Λ\curvearrowright H). As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of SL2(\mathbb Z) and its non-amenable subgroups (e.g. \mathbb Fn, with 2\leqslant n\leqslant∞) 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 SL2(\mathbb Z)\curvearrowright∏p∈ SSL2(\mathbb Zp) and SL2(\mathbb Z)\curvearrowright∏p∈ TSL2(\mathbb Zp) are neither orbit equivalent nor Borel reducible. This settles a conjecture of S. Thomas \citeTh06. 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∞ whose orbit equivalence relations have trivial fundamental group.