Orbit Equivalence Rigidity

Abstract

Consider a countable group Γ acting ergodically by measure preserving transformations on a probability space (X, µ), and let R Γ be the corresponding orbit equivalence relation on X. The following rigidity phenomenon is shown: there exist group actions such that the equivalence relation R Γ on X determines the group Γ and the action (X, µ, Γ) uniquely, up to finite groups. The natural action of SL n (Z) on the n-torus R n /Z n , for n > 2, is one of such examples. The interpretation of these results in the context of von Neumann algebras provides some support to the conjecture of Connes on rigidity of group algebras for groups with property T. Our rigidity results also give examples of countable equivalence relations of type II 1 , which cannot be generated (mod 0) by a free action of any group. This gives a negative answer to a long standing problem of Feldman and Moore.

Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II

“…On the other hand, we were not able to check Furman's condition for Bernoulli shift actions σ of higher rank lattices Γ ⊂ G, i.e. to show whether (σ, G; X) has no quotients of the form G/Λ, with Λ a lattice in G. If this would be true (which is what we expect), then (Theorem A in [Fu1]) would imply: Let G be an ICC higher rank lattice and σ Bernoulli G-action. If R Y σ ≃ R σ 0 for some Y ⊂ X and some free ergodic m.p.…”

“…He formulated the question related to his discovery that property (T) ICC groups G give rise to group factors L(G) with rigid symmetry structure ( [C1]). By now, several results in von Neumann algebras ( [CoHa], [Po8], [CSh]) and ergodic theory ( [Zi], [CoZi], [GeGo], [GoNa], [Fu1,2]) provide supporting evidence towards a positive answer to the conjecture (see V.F.R. Jones' comments on the "higher rank lattice" version of this problem in [J2]).…”

“…Note that the class of groups and actions covered by the OE rigidity results 7.6 and 7.7 is quite different from the ones in ([Fu1], [MoSh]). However, some intersection occurs with ( [MoSh]).…”

“…In fact, the terminology "strong rigidity" is borowed from ergodic theory, where "strong orbit rigidity" ( [Zi], [Fu1]), or "OE strong rigidity" ( [MoSh]), designates results showing that within a certain class of group actions orbital equivalence automatically entails isomorphism of the groups and conjugacy of the actions. While deriving the same type of conclusion, Theorem 0.1 assumes an even weaker equivalence of actions than orbit equivalence (OE), namely von Neumann equivalence (vNE), i.e.…”

“…Even more so, it provides new calculations of fundamental groups of equivalence relations, not covered by ([Ga1,2], [Fu1,2], [MoSh]). …”