Orbit inequivalent actions for groups containing a copy of  $\mathbb{F}_{2}$

Abstract

We prove that if a countable group contains a copy of F 2 , then it admits uncountably many non orbit equivalent actions. IntroductionThroughout this paper we consider free, ergodic, measure preserving (m.p.) actions (X, μ) of countable, discrete groups on standard probability spaces (X, μ). Measurable group theory is roughly the study of such group actions from the viewpoint of the induced orbit equivalence relation. A basic question in measurable group theory is to find groups which admit many non-orbit equivalent actions (see the survey [35]). In this respect, recall that two free, ergodic, m.p. actions (X, μ) and (Y, ν) are said to be orbit equivalent (OE) if they induce isomorphic equivalence relations, i.e. if there exists a measure space isomorphism θ :The striking lack of rigidity manifested by amenable groups (any two free, ergodic m.p. actions of any two infinite amenable groups and are orbit equivalent-a result proved by Dye in the case and are Abelian [8] and by Ornstein-Weiss in general ([29], see also [6])) implies that the above question is well-posed only for non-amenable groups. For a non-amenable group , it is known that admits at least two non-OE actions [5,18,36]. Moreover, recently, the following classes of non-amenable groups have been shown to admit uncountably many non-OE actions: property (T) groups [18], free groups A. Ioana ( )

Rigidity for equivalence relations on homogeneous spaces

“…In the last decade these notions of rigidity have led to several remarkable applications, most notably, to calculations of invariants of von Neumann algebras and equivalence relations ([Po06], [PV10], [Ga11]) and to constructions of non-orbit equivalent actions of non-amenable groups ( [GP05], [Io11]). …”

Topological properties of full groups

“…(It follows from the work of Gaboriau and Popa [8], Törnquist [29], and Kechris [15] that actions of the free group F 2 , up to orbit equivalence, are not classifiable by countable structures. Recently this was extended to groups containing F 2 by Ioana [13] and Kechris [15]. ) The presence of a measure allows one to define a topology on the full groups which greatly facilitates their study.…”

Countable Borel equivalence relations, Borel reducibility, and orbit equivalence

“…Theorem 2.15 (Ioana, [12]) If Γ is a countable product of a non-amenable group and an infinite amenable group, then it has continuum many orbit inequivalent measure preserving, free, ergodic actions on standard Borel probability spaces. Theorem 2.16 (Ioana, [13]) If Γ is a countable group containing F 2 as a subgroup, then it has continuum many orbit inequivalent measure preserving, free, ergodic actions on standard Borel probability spaces.…”

“…It appeared about a year or so after [13] and certain aspects of the proof were inspired by the argument given there. I am going to organize the proof somewhat differently, admittedly taking numerous shortcuts, leaving key claims unproved, and making minor simplifying assumptions.…”

“…I am going to organize the proof somewhat differently, admittedly taking numerous shortcuts, leaving key claims unproved, and making minor simplifying assumptions. The presentation I am giving will tend to obscure the links with [13].…”