On the Dimension of Group Boundaries

Abstract

The goal of this dissertation is to find connections between the small-scale dimension (i.e. covering dimension and linearly controlled dimension) of group boundaries and the large scale dimension (i.e. asymptotic dimension and macroscopic dimension) of the group. We first show that generalized group boundaries must have finite covering dimension by using finite large-scale dimension of the space. We then restrict our attention to CAT(0) group boundaries and develop metrics on the boundary that allow us to study the linearly controlled dimension. We then obtain results relating the linearly controlled dimension of CAT(0) boundaries to the large scale dimension of the CAT(0) space.