Commensurability and quasi-isometric classification of hyperbolic surface group amalgams

Let CS denote the class of groups isomorphic to the fundamental group of two closed hyperbolic surfaces identified along an essential simple closed curve in each. We construct a bi-Lipschitz map between the universal covers of these spaces equipped with a CAT(−1) metric, proving all groups in CS are quasi-isometric. The class CS has infinitely many abstract commensurability classes, which we characterize in terms of the ratio of the Euler characteristic of the two surfaces and the topological type of the curves identified. We characterize the groups in CS that contain a maximal element in their abstract commensurability class restricted to CS .