Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions

By U. Lang, Thilo Schlichenmaier. 2004

185 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6bcca30a0498e98c1146fe3f02e95850c2ab649b

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Embedding of hyperbolic groups into products of binary trees (2007, background): “the property above is fulfilled for all τ> 0. That notion is introduced in [As2], where it is called the Nagata dimension, dimN . A number of important properties and applications of dimN is obtained in [ LS ], where it is also called the Assouad–Nagata dimension.”
Dimension zero at all scales (2006, methodology): “The concept of dimension appropriate for the Lipschitz category is the Assouad-Nagata dimension [12].”
The Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current Spaces (2010, methodology): “We needed to adapt everything to integral filling volumes, so we applied a new Lipschitz extension theorem akin to that of Lang-Schlichenmaier [24].”
Hyperbolicity, CAT(−1)-spaces and the Ptolemy inequality (2006, result): “Although the proof of Theorem 1.4 needs some constant bounded away from 1 (more precisely: the proof of Theorem 1.3 in [ LS ] needs this constant), it is unknown if the result is true for any positive < 1 (similar as in the Bonk-Schramm-Assouad result).”
Dimensions of locally and asymptotically self-similar spaces (2005, methodology): “The following lemma implies in particular the finite union theorem for the capacity dimension, which is similar to the appropriate theorems for the asymptotic dimension [BD] and the Assouad-Nagata dimension [ LS ].”
Asymptotic Dimension (2007, background): “Schlichenmaier gave the following refinement of Theorem 44 [67]:”
A Hurewicz theorem for the Assouad–Nagata dimension (2006, methodology): “ongs to at least k +1−n elements of { S U′ i} k+1 i=1 . The product theorem for asymptotic dimension was proved in [15] using maps to polyhedra. The product theorem for Nagata dimension was proved in [20] using Lipschitz maps to polyhedra. Below we use Theorem 2.4 to give a simpliﬁed proof for all d”