# Rigidity in measure equivalence and orbit equivalence

Gromov's notion of Measure Equivalence of groups provides a meeting
point
between ideas from the study of $II_1$ factors and Geometric Group
Theory.
I will describe rigidity results in this framework and indicate
applications to Orbit Equivalence. In particular, there are situations
where the type $II_1$ equivalence relation induced by a group action
essentially determines the group and the action that produced it.