# On the capacity dimension of the boundary of CAT(0) spaces

## TL;DR

This paper investigates the capacity dimension of boundaries in CAT(0) spaces, comparing metrics, analyzing buildings, and proposing methods to assess their asymptotic dimensions.

## Contribution

It establishes the equivalence of visual and conical metrics for capacity dimension and explores the finiteness of asymptotic dimension in CAT(0) spaces.

## Key findings

- Visual and conical metrics yield the same capacity dimension.
- Capacity dimension of boundaries in buildings is studied.
- A method for proving finiteness of asymptotic dimension is proposed.

## Abstract

In this paper, we study the capacity dimension of the boundary of $CAT(0)$ spaces. We first compare the two metrics on the boundary of a hyperbolic $CAT(0)$ space, i.e., the visual metric and the conical metric, and prove that they 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.