| This dissertation studies certain groups by studying spaces on which they act geometrically. These spaces are studied by examining the behavior of geodesic rays in these spaces, which gives geometric data about the space that can translate into algebraic data about the group.;First, we investigate the amenability of Thompson's group F by studying the geometry of its Cayley graph. We apply the uniformly finite homology of Block and Weinberger to subsets of this graph. Many large subsets of the Cayley graph are shown to be nonamenable by exhibiting certain arrangements of geodesic rays which we call "tree-like quasi-covers".;We then examine CAT(0) boundaries. If a group acts geometrically on two CAT(0) spaces X and Y, then one obtains a G-equivariant quasi-isometry from X to Y. One may look at the image of a geodesic ray in X, and look at its closure in ∂Y. We show that this "boundary image" can have the homeomorphism type of any compact, connected subset of Euclidean space. |
