A graph product is a group constructed from a finite simple graph and a collection of groups, one for each vertex, by declaring that two vertex groups commute if their vertices are adjacent. Two well known special cases of this construction are:

Right angled Artin groups (RAAGs), when the vertex groups are all infinite cyclic.

Right angled Coxeter groups (RACGs), when the vertex groups are all cyclic of order 2. 

There is a simple graph construction that given a graph \(\Delta\) produces a graph \(\Gamma\) such that the RAAG \(A_\Delta\) is closely algebraically related to (commensurable to) the RACG \(W_\Gamma\). 

This talk will address the converse question, but from a geometric instead of algebraic perspective: Given a graph \(\Gamma\), how can one either produce a graph \(\Delta\) such that \(W_\Gamma\) is closely geometrically related to (quasiisometric to) \(A_\Delta\), or decide that no such graph exists?

This is joint work with Dani, Edletzberger, and Karrer.