A Hurewicz theorem for the Assouad-Nagata dimension

Journal of the London Mathematical Society (Impact Factor: 0.82). 06/2006; 77(3). DOI: 10.1112/jlms/jdn005
Source: arXiv


Given a function f : X → Y of metric spaces, the classical Hurewicz theorem states that dim(X) ≤ dim(f) + dim(Y). We provide analogs of this theorem
for the Assouad–Nagata dimension, asymptotic Assouad–Nagata dimension, and asymptotic dimension (the latter result generalizes
a theorem of Bell and Dranishnikov). As an application, we estimate the asymptotic Assouad–Nagata dimension of a finitely
generated group G in terms of the asymptotic Assouad–Nagata dimensions of the groups K and H from the exact sequence 1 → K → G → H → 1.

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    • "The proof is close to that of Corollary 8.5 in [5]. First, it is clear that if d H is the Hausdorff metric, then the projection map f : (G, d G ) → (H, d H ) "
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    • "Such dimension can be considered as the linear version of the asymptotic dimension. In recent years a part of the research activity was focused on this dimension and its relationship with the asymptotic dimension (see for example [16], [9], [10], [3], [4], [6], [5], [17] [12], [15]). One of the main problems of interest consists in studying the differences between the asymptotic dimension and the asymptotic Assouad-Nagata dimension in the context of the geometric group theory. "
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    • "Applying 2.7 we deduce asdim AN (G (n,k) , d (n,k) ) ≥ n + k. The other inequalities follow easily from the by the subadditivity of the asymptotic dimension and the Assouad-Nagata dimension with respect to the cartesian product(see for example [5]) wand the well known fact asdim(Z n , d 1 ) = n. Problem 4.12. "
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    ABSTRACT: In this work we study two problems about Assouad-Nagata dimension: 1) Is there a metric space of non zero Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes) 2) Suppose $G$ is a locally finite group with a proper left invariant metric $d_G$. If $\dim_{AN}(G, d_G)>0$, is $\dim_{AN} (G, d_G)$ infinite?(Brodskiy, Dydak and Lang) The first question is answered positively not only for general metric spaces but also for discrete groups with proper left invariant metrics. The second question has a negative solution. We show that for each $n$ there exists a locally finite group of Assouad-Nagata dimension $n$. A generalization to countable groups of arbitrary asymptotic dimension is given
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