On the capacity dimension of the boundary of cat(0) spaces

In this dissertation, we study the capacity dimension of the boundary of CAT(0) spaces. We believe the capacity dimension of the boundary of space is useful in understanding the large-scale geometry of space, such as the asymptotic dimension. More specifically, we first compare the two metrics on the boundary of a hyperbolic CAT(0) space, i.e., the visual metric and Moran's metric, and conclude that the two metrics give the same capacity dimension of the boundary. Then we study the capacity dimension of the boundary of buildings, which is an important class of CAT(0) spaces. Finally, we give a possible method to prove the finiteness of the asymptotic dimension of CAT(0) spaces.