  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,Gamma)
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, 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.
