Orbit equivalence rigidity

Consider a countable group G acting ergodically by
measure preserving transformations on a probability space (X,m), and let
R_G be the corresponding orbit equivalence relation on X. We show a new
rigidity phenomenon in the context of orbit equivalence: there exist group
actions such that the equivalence relation R_G on X determines the group
G and the action (X,m,G) 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. We also give a list of 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.

Bibliographical: Ann. of Math. (2)

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[Papers] [Alex Furman]