Topics in geometric group theory

This document contains results in a couple of nonrelated areas of geometric group theory. What follows are abstracts for each part. Let Mi and Ni be path-connected locally uniquely geodesic metric spaces that are not points and f : ?mi=1Mi ? ?ni=1 Ni be an isometry where ?ni=1 Ni and ?mi=1Mi are given the sup metric. Then m = n and after reindexing Mi is isometric to Ni for all i. Moreover f is a composition of an isometry that reindexes the factor spaces and an isometry that is a product of isometries fi : Mi ? Ni. Given a geometric amalgamation of free groups G and the associated simple thick two-dimensional hyperbolic piecewise manifold M, the visual boundary ?M is a complete quasi-isometry invariant. This invariant can be effciently computed for any G using an adaptation of Leighton's Theorem. Let G and G0 be geometric amalgamation of free groups with a single Z vertex. If the associated simple thick two-dimensional hyperbolic piecewise manifolds M and M0 have the same Euler characteristic, then G is commensurable to G0 if and only if M and M0 are homeomorphic. The proof is then extended to the case where G and G0 have more than a single Z vertex, but more conditions have to be placed on G and G0. With these results an elementary example of two geometric amalgamations of free groups that are quasi-isometric but not commensurable can be given.