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

Adrian Ioana

Journal of the European Mathematical Society, Tome 18 (2016) no. 12, pp. 2733-2784

Résumé

We study equivalence relations R(Γ↷G) that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action Γ↷G is free and has spectral gap, we describe precisely when R(Γ↷G) is orbit equivalent or Borel reducible to another such equivalence relation R(Λ↷H). As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of SL2​(Z) and its non-amenable subgroups (e.g. Fn​, with 2⩽n⩽∞) 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​(Z)↷∏p∈S​SL2​(Zp​) and SL2​(Z)↷∏p∈T​SL2​(Zp​) are neither orbit equivalent nor Borel reducible. This settles a conjecture of S. Thomas [Th01,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 F∞​ whose orbit equivalence relations have trivial fundamental group.

Détail

DOI : 10.4171/jems/652

Classification : 37-XX, 46-XX
Keywords: Spectral gap, rigidity, orbit equivalence, Borel reducibility, equivalence relations, profinite actions, outer automorphism group, II1​ factor