Weak amenability of free products of hyperbolic and amenable groups

TL;DR

This paper proves that the free product of an amenable group and a hyperbolic group is weakly amenable, using orbit equivalence to relate it to a simpler free product involving the integers.

Contribution

It establishes the weak amenability of free products of amenable and hyperbolic groups, a new result in geometric group theory.

Findings

Free product of an amenable and a hyperbolic group is weakly amenable.

The proof utilizes orbit equivalence to relate the free product to a simpler case.

Provides new insights into the structure of free products in geometric group theory.

Abstract

We show that, if $G$ is an amenable group and $H$ is a hyperbolic group, then the free product $G\ast H$ is weakly amenable. A key ingredient in the proof is the fact that $G\ast H$ is orbit equivalent to $\mathbb{Z}\ast H$.