Geodesic spaces of low Nagata dimension

TL;DR

This paper establishes that geodesic metric spaces with certain planar embeddings have Nagata dimension at most two, and extends the result to three-dimensional Hadamard manifolds, impacting their Lipschitz retract properties.

Contribution

It proves that geodesic spaces with planar embeddings have Nagata dimension at most two, answering a recent open question and extending to Hadamard manifolds.

Findings

Geodesic spaces with planar embeddings have Nagata dimension ≤ 2

Planar graphs have Nagata dimension ≤ 2

Three-dimensional Hadamard manifolds have Nagata dimension 3

Abstract

We show that every geodesic metric space admitting an injective continuous map into the plane as well as every planar graph has Nagata dimension at most two, hence asymptotic dimension at most two. This relies on and answers a question in a very recent work by Fujiwara and Papasoglu. We conclude that all three-dimensional Hadamard manifolds have Nagata dimension three. As a consequence, all such manifolds are absolute Lipschitz retracts.