Hurewicz Theorem for Assouad-Nagata dimension

Abstract

Given a function $f:X\to Y$ of metric spaces, its {\it asymptotic dimension} $\asdim(f)$ is the supremum of $\asdim(A)$ such that $A\subset X$ and $\asdim(f(A))=0$. Our main result is \begin{Thm} \label{ThmAInAbstract} $\asdim(X)\leq \asdim(f)+\asdim(Y)$ for any large scale uniform function $f:X\to Y$. \end{Thm} \ref{ThmAInAbstract} generalizes a result of Bell and Dranishnikov in which $f$ is Lipschitz and $X$ is geodesic. We provide analogs of \ref{ThmAInAbstract} for Assouad-Nagata dimension ${\dim }_{AN}$ and asymptotic Assouad-Nagata dimension $\ANasdim$. In case of linearly controlled asymptotic dimension $\Lasdim$ we provide counterexamples to three questions in a list of problems of Dranishnikov. As an application of analogs of \ref{ThmAInAbstract} we prove \begin{Thm} \label{ThmBInAbstract} If $1\to K\to G\to H\to 1$ is an exact sequence of groups and $G$ is finitely generated, then $$\ANasdim (G,d_G)\leq \ANasdim (K,d_G|K)+\ANasdim (H,d_H)$$ for any word metrics metrics $d_G$ on $G$ and $d_H$ on $H$. \end{Thm} \ref{ThmBInAbstract} extends a result of Bell and Dranishnikov for asymptotic dimension.

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• Asymptotic topology
• Every Coxeter group acts amenably on a compact space
• Asymptotic dimension in Bedlewo
• Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions
• A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory
• Dimension of locally and asymptotically self-similar spaces
• Nagata-Assouad dimension via Lipschitz extensions
• Coarse dimensions and partitions of unity

Cited by in corpus (27)

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• Assouad-Nagata dimension of tree-graded spaces
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• Asymptotic dimension of planes and planar graphs
• Coarse geometry of quasi-transitive graphs beyond planarity
• On the dynamic asymptotic dimension of étale groupoids
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• On stability of asymptotic property C for products and some group extensions
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• Assouad-Nagata dimension of finitely generated C'(1/6) groups
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• Dimension-Raising Maps in a Large Scale
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• Lipschitz extension theorems with explicit constants
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• The Hausdorff- and Nagata-dimension of Möbius spaces