Damien Gaboriau

An uncountable family of nonorbit equivalent actions of _

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

Orbit Equivalence and Measured Group Theory

We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions.

Von Neumann Algebras and Ergodic Theory of Group Actions

The theory of von Neumann algebras has seen some dramatic advances in the last few years. Von Neumann algebras are objects which can capture and analyze symmetries of mathematical or physical situations whenever these symmetries can be cast in terms of generalized morphisms of the algebra (Hilbert bimodules, or correspondences). Analyzing these symmetries led to an amazing wealth of new mathematics and the solution of several long-standing problems in the theory. Popa’s new deformation and rigidity theory has culminated in the discovery of new cocycle superrigidity results à la Zimmer, thus establishing a new link to orbit equivalence ergodic theory. The workshop brought together world-class researchers in von Neumann algebras and ergodic theory to focus on these recent developments. Mathematics Subject Classification (2000): 46L10. Introduction by the Organisers The workshop Von Neumann Algebras and Ergodic Theory of Group Actions was organized by Dietmar Bisch (Vanderbilt University, Nashville), Damien Gaboriau (ENS Lyon), Vaughan Jones (UC Berkeley) and Sorin Popa (UC Los Angeles). It was held in Oberwolfach from October 26 to November 1, 2008. This workshop was the first Oberwolfach meeting on von Neumann algebras and orbit equivalence ergodic theory. The organizers took special care to invite many young mathematicians and more than half of the 28 talks were given by them. The meeting was very well attended by over 40 participants, leading senior researchers and junior mathematicians in the field alike. Participants came from 2764 Oberwolfach Report 49/2008 about a dozen different countries including Belgium, Canada, Denmark, France, Germany, Great Britain, Japan, Poland, Switzerland and the USA. The first day of the workshop featured beautiful introductory talks to orbit equivalence and von Neumann algebras (Gaboriau), Popa’s deformation/rigidity techniques and applications to rigidity in II1 factors (Vaes), subfactors and planar algebras (Bisch), random matrices, free probability and subfactors (Shlyakhtenko), subfactor lattices and conformal field theory (Xu) and an open problem session (Popa). There were many excellent lectures during the subsequent days of the conference and many new results were presented, some for the first time during this meeting. A few of the highlights of the workshop were Vaes’ report on a new cocycle superrigidity result for non-singular actions of lattices in SL(n,R) on Rn and on other homogeneous spaces (joint with Popa), Ioana’s result showing that every sub-equivalence relation of the equivalence relation arising from the standard SL(2,Z)-action on the 2-torus T is either hyperfinite, or has relative property (T), and Epstein’s report on her result that every countable, non-amenable group admits continuum many non-orbit equivalent, free, measure preserving, ergodic actions on a standard probability space. Other talks discussed new results on fundamental groups of II1 factors, L -rigidity in von Neumann algebras, II1 factors with at most one Cartan subalgebra, subfactors from Hadamard matrices, a new construction of subfactors from a planar algebra and new results on topological rigidity and the Atiyah conjecture. Many interactions and stimulating discussions took place at this workshop, which is of course exactly what the organizers had intended. The organizers would like to thank the Mathematisches Forschungsinstitut Oberwolfach for providing the splendid environnment for holding this conference. Special thanks go to the very helpful and competent staff of the institute.