Boundary Amenability of Relatively Hyperbolic Groups

TL;DR

This paper proves that relatively hyperbolic groups are exact if their peripheral groups are, by demonstrating group actions on compact spaces are amenable, with implications for von Neumann algebras and orbit equivalence.

Contribution

It establishes the boundary amenability of relatively hyperbolic groups based on the exactness of their peripheral groups, extending understanding of group actions.

Findings

Relatively hyperbolic groups are exact if all peripheral groups are exact.

Groups acting on fine hyperbolic graphs with finite quotient are exact under certain conditions.

Applications to group von Neumann algebras and measurable orbit equivalence relations.

Abstract

Let K be a fine hyperbolic graph and G be a group acting on K with finite quotient. We prove that G is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group G acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations.