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Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. (English) Zbl 1095.53033
A variant of the Gromov’s notion of asymptotic dimension is discussed in this paper with a view towards applications in analysis on metric spaces. Suppose that \(X=(X,d)\) is a metric space and \(\mathcal B=(B_i)_{i\in I}\) is a family of subsets of \(X\). The family \(\mathcal B\) is called \(D\)-bounded, for some constant \(D\geq 0\), if \(\text{diam\,} B_i:= \sup\{ d(x,x') : x,x'\in B_i \}\leq D\) for all \(i\in I\). The multiplicity of the family is the infimum of all integers \(n\geq 0\) such that each point in \(X\) belongs to no more than \(n\) members of \(\mathcal B\). For \(s>0\) the \(s\)-multiplicity of \(\mathcal B\) is the infimum of all \(n\) such that every subset of \(X\) with diameter \(\leq s\) meets at most \(n\) members of the family. The asymptotic dimension, \(\text{asdim}\,X\), of \(X\) is defined as the infimum of all integers \(n\) such that for every \(s>0\), \(X\) possesses a \(D\)-bounded covering with \(s\)-multiplicity at most \(n+1\) for some \(D=D(s)<\infty\). The Nagata dimension, or Assouad-Nagata dimension, \(\dim_N X\), of \(X\) is the infimum of all integers \(n\) with the following property: there exists a constant \(c>0\) such that for all \(s>0\), \(X\) has a \(cs\)-bounded covering with \(s\)-multiplicity at most \(n+1\). Clearly \(\dim_N X\geq \text{asdim} X\). The number \(\dim_N X\) is unaffected if the covering sets are required to be open, or closed, or if the “test set” with diameter \(\leq s\) is replaced by an open or closed ball of radius \(s\).
This Nagata dimension was introuced by P. Assouad [C.R. Acad. Sci. Paris Sér. I Math. 294, No. 1, 31–34 (1982; Zbl 0481.54015)]; it is closely related to a theorem of J. Nagata [Fund. Math. 45, 143–181 (1958; Zbl 0086.15802)] characterizing the topological dimension of metrizable spaces. In contrast to the asymptotic dimension, the Nagata dimension of a metric space is in general not preserved under quasi-isometries, but it is still a bi-Lipschitz invariant and, as it turns out, even a quasisymmetry invariant (cf. Theorem 1.2). The class of metric spaces with finite Nagata dimension includes all doubling spaces, geodesic Gromov hyperbolic spaces that are doubling in the small, metric (\(\mathbb R\)-)trees, Euclidean buildings, and homogeneous Hadamard manifolds, among others. Moreover, this class is closed under taking finite products and finite unions.
One of the main results interposes between theorems of P. Assouad [Bull. Soc. Math. Fr. 111, No. 4, 429–428 (1983; Zbl 0597.54015)] and A. N. Dranishnikov [Trans. Am. Math. Soc. 355, No. 1, 155–167 (2003; Zbl 1020.53025)]: every metric space with Nagata dimension at most \(n\) admits a quasisymmetric embedding into the product of \(n+1\) metric trees (cf. Theorem 1.3). Another circle of results relates the concept of extendability of partially defined Lipschitz maps between metric spaces. Two general extension theorems are obtained involving a Lipschitz connectedness assumption on the target space and a bound on the Nagata dimension of either the domain of the given map or its complement (cf. Theorems 1.5 and 1.6).

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
54E35 Metric spaces, metrizability
54F45 Dimension theory in general topology
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