A Comparison of Large Scale Dimension of a Metric Space to the Dimension of its Boundary

TL;DR

This paper explores the relationship between the large-scale dimension of hyperbolic groups and the topological dimension of their boundaries, extending known results to groups with $ ext{Z}$-structures.

Contribution

It extends the known equality between asymptotic dimension and boundary dimension from hyperbolic groups to all groups with $ ext{Z}$-structures, providing a lower bound.

Findings

Asymptotic dimension of groups with $ ext{Z}$-structures is at least the boundary dimension plus one.

Partial extension of Buyalo and Lebedeva's result to a broader class of groups.

Abstract

Buyalo and Lebedeva have shown that the asymptotic dimension of a hyperbolic group is equal to the dimension of the group boundary plus one. Among the work presented here is a partial extension of that result to all groups admitting $\mathcal{Z}$-structures; in particular, we show that $\hbox{asdim}G\geq \hbox{dim}Z+1$ where $Z$ is the $\mathcal{Z}$-boundary.