Equivalence relations that act on bundles of hyperbolic spaces

TL;DR

This paper studies how measured equivalence relations act on bundles of hyperbolic spaces, establishing uniqueness of maximal hyperfinite subrelations, classifying group elements, and extending Tits' alternative.

Contribution

It introduces a classification of elements based on boundary actions and proves the uniqueness of maximal hyperfinite subequivalence relations in this setting.

Findings

Every aperiodic hyperfinite subequivalence relation is contained in a unique maximal hyperfinite subrelation.

Classifies elements of the full group according to boundary measure actions.

Establishes an analogue of Tits' alternative for these relations.

Abstract

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.