Asymptotic dimension, Property A, and Lipschitz maps

TL;DR

This paper explores the relationship between asymptotic dimension, Property A, and Lipschitz maps, establishing a coarse geometric analogue of classical topological dimension theory.

Contribution

It introduces a new coarse dimension concept based on asymptotically Lipschitz maps, linking it to Property A and Gromov's asymptotic dimension.

Findings

Asymptotic dimension coincides with Gromov's asymptotic dimension.

Property A spaces relate to coarse paracompactness.

Analogies between classical and coarse dimension theories are established.

Abstract

It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns out the analog of maps f from X to K is related to asymptotically Lipschitz maps, the analog of paracompact spaces are spaces related to Yu's Property A, and the dimension coincides with Gromov's asymptotic dimension.