Commensurability classification of certain Coxeter groups and amalgams of free groups

Abstract

Two groups $G$ and $H$ are (abstractly) commensurable if they have finite index subgroups $G^{\prime }$ and $H^{\prime }$ so that $G^{\prime }$ and $H^{\prime }$ are abstractly isomorphic. Commensurability in this sense is an equivalence relation on abstract groups. We give explicit necessary and sufficient conditions for commensurability within certain families of Coxeter groups and amalgams of free groups. Our methods are topological, and involve a new geometric realisation of the Coxeter groups involved. This is joint work with Pallavi Dani and Emily Stark.