  We study rigidity properties of lattices in hyperbolic n-space with n>2 and
of surface groups in the context of (integrable) measure equivalence. The
results for lattices in hyperbolic n-space with n>2 are generalizations of
Mostow rigidity; they include a cocycle version of strong rigidity and an
integrable measure equivalence classification. Despite the lack of Mostow
rigidity for n=2 we show that cocompact lattices in SL(2,R) allow a similar
integrable measure equivalence classification.
