A horospherical ratio ergodic theorem for actions of free groups

Abstract

We prove a ratio ergodic theorem for discrete non-singular measurable equivalence relations, provided they satisfy a strong form of the Besicovich covering property. In particular, this includes all hyperfinite measurable equivalence relation. We then use this result to study general non-singular actions of non-abelian free groups and establish a ratio ergodic theorem for averages along horospheres.