  Consider a measured equivalence relation acting on a bundle of hyperbolic
metric spaces by isometries. We prove that every aperiodic hyperfinite
subequivalence relation is contained in a {\em unique} maximal hyperfinite
subequivalence relation. We classify elements of the full group according to
their action on fields on boundary measures (extending earlier results of
Kaimanovich), study the existence and residuality of different types of
elements and obtain an analogue of Tits' alternative.
