PreprintCommensurability for certain rightangled Coxeter groups and geometric amalgams of free groupsPallavi Dani, Emily Stark and Anne ThomasAbstractWe give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1ended, hyperbolic groups, namely rightangled Coxeter groups defined by generalized thetagraphs and cycles of generalized thetagraphs, 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 rightangled Coxeter group, and give an example of a geometric amalgam of free groups which is not commensurable to any group which is finitely generated by torsion elements. Our proofs involve a new geometric realization of the rightangled Coxeter groups we consider, such that covers corresponding to torsionfree, finiteindex subgroups are surface amalgams. This paper is available as a pdf (856kB) file.
