  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.
