Equivalence relations that act on bundles of hyperbolic spaces

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 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 [Boundary amenability of hyperbolic spaces.Discrete Geometric Analysis(Contemporary Mathematics, 347). American Mathematical Society, Providence, RI, 2004, pp. 83–111]), study the existence and residuality of different types of elements and obtain an analog of Tits’ alternative.