Asymptotic topological regularity of CAT(0) spaces

TL;DR

This paper investigates the large-scale topological structure of CAT(0) spaces, showing that those with small volume growth are topologically equivalent to Euclidean space, thus linking geometric growth conditions to topology.

Contribution

It establishes a new topological regularity result for proper, geodesically complete CAT(0) spaces based on volume growth constraints.

Findings

Spaces with small volume growth are homeomorphic to Euclidean space.

The paper characterizes the asymptotic geometry of these CAT(0) spaces.

Provides conditions under which CAT(0) spaces exhibit Euclidean topology.

Abstract

We study asymptotic topological regularity of CAT(0) spaces. We prove that if a purely n-dimensional, proper, geodesically complete CAT(0) space has small volume growth, then it is homeomorphic to the n-dimensional Euclidean space. We also discuss asymptotic geometry of proper, geodesically complete CAT(0) spaces of small volume growth.