KAZHDAN GROUPS, COCYCLES AND TREES

We study cocycles of Kazhdan group actions with values in groups acting on trees. In particular we show that all cocycles of finite measure preserving actions of Kazhdan groups taking values in a free group are cohomologically trivial. Analogous techniques show that a Kazhdan equivalence relation is not treeable. 1. Introduction. Let G be a locally compact group. Recall (Zi) that a unitary representation p of G on a Hilbert space We almost has invariant vectors if for all E > 0 and compact subsets K of G there is a unit vector v E We such that 11p(g)v - vllx < E for all g E K. We say that G has Kazhdan's property T or that G is a Kazhdan group if all unitary representations of G that almost have invariant vectors actually