ATmenability of some graphs of groups with cyclic edge groups

TL;DR

This paper proves that specific graphs of groups with cyclic edge groups are aTmenable, especially when vertex groups are virtually special or act properly on hyperbolic space.

Contribution

It establishes aTmenability for graphs of groups with cyclic edge groups under new conditions involving vertex group properties.

Findings

Graphs of groups with cyclic edge groups are aTmenable under certain conditions.

Vertex groups being virtually special or acting properly on hyperbolic space ensures aTmenability.

The results extend understanding of aTmenability in geometric group theory.

Abstract

We show that certain graphs of groups with cyclic edge groups are aTmenable. In particular, this holds when each vertex group is either virtually special or acts properly and semisimply on $\mathbb{H}^n$.