Metrics on Visual Boundaries of CAT(0) Spaces

TL;DR

This paper investigates the relationship between the large-scale and small-scale dimensions of CAT(0) groups and their boundaries, proposing new metrics and establishing bounds on dimensions to advance understanding of their geometric properties.

Contribution

It introduces two new metrics for CAT(0) boundaries, analyzes their properties, and links boundary dimensions to the group's macroscopic dimension, addressing open problems in geometric group theory.

Findings

Linearly controlled dimension of the boundary remains finite under one metric.

Macroscopic dimension of the group is bounded by twice the boundary's linearly controlled dimension plus one.

Basic examples and open questions are discussed.

Abstract

A famous open problem asks whether the asymptotic dimension of a CAT(0) group is necessarily finite. For hyperbolic groups, it is known that asymptotic dimension of the group is bounded above by the dimension of the boundary plus one, which is known to be finite. For CAT(0) groups, the latter quantity is also known to be finite, so one approach is to try proving a similar inequality. So far those efforts have failed. Motivated by these questions we work toward understanding the relationship between large scale dimension of CAT(0) groups and small scale dimension of the group's boundary by shifting attention to the linearly controlled dimension of the boundary. To do that, one must choose appropriate metrics for the boundaries. In this paper, we suggest two candidates and develop some basic properties. Under one choice, we show that linearly controlled dimension of the boundary remains finite; under another choice, we prove that macroscopic dimension of the group is bounded above by twice the linearly controlled dimension of the boundary plus one. Other useful results are established, some basic examples are analyzed, and a variety of open questions are posed.