Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups

TL;DR

This paper establishes explicit criteria for when certain hyperbolic groups, including right-angled Coxeter groups and geometric amalgams of free groups, are commensurable, revealing deep geometric and algebraic relationships.

Contribution

It provides necessary and sufficient conditions for commensurability among these groups and shows their interrelations through new geometric realizations and JSJ graph analysis.

Findings

Criteria for commensurability of right-angled Coxeter groups and geometric amalgams.

All geometric amalgams with JSJ graph as a tree are commensurable to right-angled Coxeter groups.

Example of a geometric amalgam not quasi-isometric to any finitely generated torsion group.

Abstract

We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized theta-graphs and cycles of generalized theta-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of diameter at most 4. We also show that if a geometric amalgam of free groups has JSJ graph a tree, then it is commensurable to a right-angled Coxeter group, and give an example of a geometric amalgam of free groups which is not quasi-isometric (hence not commensurable) to any group which is finitely generated by torsion elements. Our proofs involve a new geometric realization of the right-angled Coxeter groups we consider, such that covers corresponding to torsion-free, finite-index subgroups are surface amalgams.