A Hurewicz theorem for the Assouad-Nagata dimension

Journal of the London Mathematical Society (Impact Factor: 0.88). 06/2006; DOI: 10.1112/jlms/jdn005
Source: arXiv

ABSTRACT 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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Available from: J. Dydak, Jul 10, 2015
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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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