Orbit equivalence rigidity

• 成果类型: Article
• 署名作者: Furman, A
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/121063
• 发表日期: 1999
• 页码: 1083-1108
• 关键词: ergodic actions; Lie group

摘要

Consider a countable group Gamma acting ergodically by measure preserving transformations on a probability space (X, mu), and let R-Gamma be the corresponding orbit equivalence relation on X. The following rigidity phenomenon is shown: there exist group actions such that the equivalence relation R-Gamma on X determines the group Gamma and the action (X, mu, r) uniquely, up to finite groups. The natural action of SLn(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 II1, 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.