Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions

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.

References in corpus (1)

• Universal spaces for asymptotic dimension

Cited by in corpus (10)

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• Dimension of locally and asymptotically self-similar spaces
• Nagata-Assouad dimension via Lipschitz extensions
• Coarse geometry of quasi-transitive graphs beyond planarity
• Assouad-Nagata dimension of locally finite groups and asymptotic cones
• Capacity dimension and embedding of hyperbolic spaces into the product of trees
• Erratum: Lipschitz connectivity and filling invariants in solvable groups and buildings