Article

A Hurewicz theorem for the Assouad-Nagata dimension

(Impact Factor: 0.82). 06/2006; 77(3). 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.

Full-text

Available from: J. Dydak, Oct 09, 2015
0 Followers
·
• Source
• "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 ) "
Article: Assouad-Nagata dimension of connected Lie groups
[Hide abstract]
ABSTRACT: We prove that the asymptotic Assouad-Nagata dimension of a connected Lie group \$G\$ equipped with a left-invariant Riemannian metric coincides with its topological dimension of \$G/C\$ where \$C\$ is a maximal compact subgroup. To prove it we will compute the Assouad-Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad-Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometric to any cocompact lattice on a connected Lie group. Comment: 21 pages. Main theorem has been extended to connected Lie groups. Added section 6, section 7 and example 4.11
Mathematische Zeitschrift 10/2009; 273(1-2). DOI:10.1007/s00209-012-1004-1 · 0.69 Impact Factor
• Source
• "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. "
Article: Assuad-Nagata dimension of nilpotent groups with arbitrary left invariant metrics
[Hide abstract]
ABSTRACT: Suppose \$G\$ is a countable, not necessarily finitely generated, group. We show \$G\$ admits a proper, left-invariant metric \$d_G\$ such that the Assouad-Nagata dimension of \$(G,d_G)\$ is infinite, provided the center of \$G\$ is not locally finite. As a corollary we solve two problems of A.Dranishnikov.
Proceedings of the American Mathematical Society 05/2008; 138(6). DOI:10.1090/S0002-9939-10-10240-8 · 0.68 Impact Factor
• Source
• "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. "
Article: Assouad-Nagata dimension of locally finite groups and asymptotic cones
[Hide abstract]
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
Topology and its Applications 12/2007; 157(17). DOI:10.1016/j.topol.2010.07.015 · 0.55 Impact Factor