Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions

Details Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions

2004-10-04

arxiv.org/abs/math/0410048v1

Mathematics

Metric Geometry

Scientific paper

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.

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