Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions

We investigate a variant of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. This turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes doubling spaces, Gromov hyperbolic spaces of bounded local geometry, Euclidean buildings, and homogeneous Hadamard manifolds, among others, and is closed under taking finite products and finite unions. We prove a quasisymmetric embedding theorem and two comprehensive Lipschitz extension theorems involving a bound on the Nagata dimension.