Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups (Q2418868)

Summary: 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 \(\leq 4\). We also show that if a geometric amalgamof 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.