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.
Cite this article
Lewis Bowen, Amos Nevo, A horospherical ratio ergodic theorem for actions of free groups. Groups Geom. Dyn. 8 (2014), no. 2, pp. 331–353DOI 10.4171/GGD/228