Strong Rigidity for Ergodic Actions of Semisimple Lie Groups

Abstract

The aim of this paper is to prove an analogue of the strong rigidity theorems for lattices in semisimple Lie groups of Mostow and Margulis in the context of ergodic actions of semisimple Lie groups and ergodic foliations by symmetric spaces. If G is a locally compact group and S is an ergodic G-space with finite invariant measure, one can attempt to study the action by means of the equivalence relation on S which it defines. Two actions (of possibly different groups) are called orbit equivalent if they define isomorphic equivalence relations. Of course for two actions of the same group, conjugacy of the actions (or more generally, automorphic conjugacy, i.e., conjugacy modulo an automorphism of the group) implies orbit equivalence. However in general, orbit equivalence is, to a surprising extent, a far weaker notion. This