Bi-exactness of relatively hyperbolic groups

TL;DR

This paper establishes that finitely generated relatively hyperbolic groups are bi-exact precisely when their peripheral subgroups are bi-exact, extending Ozawa's result from amenability to bi-exactness.

Contribution

It generalizes Ozawa's theorem by replacing amenability with bi-exactness for peripheral subgroups in relatively hyperbolic groups.

Findings

Finitely generated relatively hyperbolic groups are bi-exact iff all peripheral subgroups are bi-exact.

Extends Ozawa's result from amenability to bi-exactness.

Provides a characterization linking peripheral subgroup properties to the bi-exactness of the entire group.

Abstract

We prove that finitely generated relatively hyperbolic groups are bi-exact if and only if all peripheral subgroups are bi-exact. This is a generalization of Ozawa's result which claims that finitely generated relatively hyperbolic groups are bi-exact if all peripheral subgroups are amenable.