Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions

TL;DR

This paper explores Nagata dimension, a quasisymmetry invariant of metric spaces, demonstrating its properties, including embeddings and Lipschitz retracts, with implications for various classes of geometric spaces.

Contribution

It introduces a quasisymmetric embedding theorem for spaces with finite Nagata dimension and characterizes absolute Lipschitz retracts within this class.

Findings

Finite Nagata dimension is a quasisymmetry invariant.

Spaces with finite Nagata dimension include doubling spaces, trees, and certain manifolds.

Established a quasisymmetric embedding theorem for these spaces.

Abstract

We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes in particular all doubling spaces, metric trees, euclidean buildings, and homogeneous or pinched negatively curved Hadamard manifolds. Among others, we prove a quasisymmetric embedding theorem for spaces with finite Nagata dimension in the spirit of theorems of Assouad and Dranishnikov, and we characterize absolute Lipschitz retracts of finite Nagata dimension.