AN UNCOUNTABLE FAMILY OF NONORBIT EQUIVALENT ACTIONS OF Fn

Recall that two ergodic probability measure preserving (p.m.p.) actions σi for i = 1, 2 of two countable groups Γi on probability measure standard Borel spaces (Xi, μi) are orbit equivalent (OE) if they define partitions of the spaces into orbits that are isomorphic, more precisely, if there exists a measurable, almost everywhere defined isomorphism f : X1 → X2 such that f∗μ1 = μ2 and the Γ1-orbit of μ1almost every x ∈ X1 is sent by f onto the Γ2-orbit of f(x). The theory of orbit equivalence, although underlying the “group measure space construction” of Murray and von Neumann [MvN36], was born with the work of H. Dye who proved, for example, the following striking result [Dy59]: Any two ergodic p.m.p. free actions of Γ1 Z and Γ2 ⊕ j∈N Z/2Z are orbit equivalent. Through a series of works, the class of groups Γ2 satisfying Dye’s theorem gradually increased until it achieved the necessary and sufficient condition: Γ2 is infinite amenable [OW80]. In particular, all infinite amenable groups produce one and only one ergodic p.m.p. free action up to orbit equivalence (see also [CFW81] for a more general statement). By using the notion of strong ergodicity, A. Connes and B. Weiss proved that any nonamenable group without Kazhdan property (T) admits at least two non-OE p.m.p. free ergodic actions [CW80]. The first examples of groups with uncountably many non-OE free ergodic actions was obtained in [BG81], using a somewhat circumstantial construction, based on prior work in [McD69]. Within the circle of ideas brought up by Zimmer’s cocycle super-rigidity [Zi84], certain Kazhdan property (T) lattices of Lie groups such as SL(n,Z), n ≥ 3, were shown to admit uncountably many non-OE free ergodic actions as well (see [GG88]). It is only recently that we learned that this property was shared by all infinite groups with Kazhdan property (T) [Hj02-b], and thanks to another reason (bounded cohomology) by many torsion free finitely generated direct products Γ1×Γ2, including nontrivial (l ≥ 2) products of free groups like Fp1 × Fp2 × · · · × Fpl , pi ≥ 2 [MS02]. On the other hand, the situation for the free groups themselves or SL(2,Z) remained unclear and arouse the interest of producing more non-OE free ergodic actions of Fn than just the two given by [CW80] (more precisely in producing ways