Nagata-Assouad dimension via Lipschitz extensions

TL;DR

This paper explores the relationships between various metric space dimensions and Nagata-Assouad dimension using Lipschitz extensions, providing new characterizations and scale analogs.

Contribution

It introduces functors relating different dimension theories to Nagata-Assouad dimension via Lipschitz categories and characterizes spaces with finite Nagata-Assouad dimension through Lipschitz extensibility of spheres.

Findings

Relates asymptotic and capacity dimensions to Nagata-Assouad dimension.

Identifies spaces with finite Nagata-Assouad dimension via Lipschitz extension properties.

Provides large-scale and small-scale analogs of the main results.

Abstract

In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov, and capacity dimension of Buyalo \cite{Buyalo1}) to Nagata-Assouad dimension. This is done by applying two functors on the Lipschitz category of metric spaces: microscopic and macroscopic. In the second part we identify (among spaces of finite Nagata-Assouad dimension) spaces of Nagata-Assouad dimension at most $n$ as those for which the $n$-sphere $S^n$ is a Lipschitz extensor. Large scale and small scale analogs of that result are given.