The impact of Pontryagin and Bohr functors on large-scale properties of locally compact abelian groups
type: Item
instance of: scholarly article
title: The impact of Pontryagin and Bohr functors on large-scale properties of locally compact abelian groups
zbMATH Open document ID: 1439.22010
author: Dikran Dikranjan
author: Nicolò Zava
published in: Topology and its Applications
publication date: 2020-04-07T00:00:00Z
review text: The theory of coarse geometry found applications in several branches of mathematics. The authors of the paper under review are interested of coarse structures on topological groups. An outline of coarse groups from categorical point of view was given by \textit{D. Dikranjan} and \textit{N. Zava} [Topology Appl. 273, Article ID 106980, 31 p. (2020; Zbl 1442.18025)]. One of the most important coarse invariant is the asymptotic dimension where was introduced by Gromov. A significant part of the article is devoted to the study of an asymptotic dimension. For example, the authors prove that if \(H\) be a closed subgroup of a locally compact abelian group \(G\) then \(\mathrm{asdim}\; G=\mathrm{asdim} \;G/H+\mathrm{asdim} \;H\) and \(\mathrm{asdim}\;G = n_\mathbb{R}(G) + \varrho_0(G)\), where \(n_\mathbb{R}(G)\) and \(\varrho_0(G)\) are simple cardinal invariants of \(G\). The authors provide a characterisation of locally compact groups with asymptotic dimension \(0\). If \(f : H\rightarrow G\) is an epimorphism in \textbf{LCA} then \(\mathrm{asdim}\;H \geq\mathrm{asdim}\;G\). In the last section of the paper, the authors study the impact of the Bohr functor on the compact-group coarse structure with particular emphasis on the preservation of the asymptotic dimension. Some open questions are also contained.
Mathematics Subject Classification ID: 22B05
Mathematics Subject Classification ID: 18A99
Mathematics Subject Classification ID: 20K45
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 7187672
zbMATH Keywords: functorial coarse structure
zbMATH Keywords: coarse group
zbMATH Keywords: group ideal
zbMATH Keywords: asymptotic dimension
zbMATH Keywords: covering dimension
zbMATH Keywords: Pontryagin duality
zbMATH Keywords: Bohr topology
zbMATH Keywords: coarse space
zbMATH Keywords: coarse equivalence
reviewed by: Nikolay I. Kryuchkov
MaRDI profile type: Publication
full work available at URL: j.topol.2019.107025
OpenAlex ID: W2997779757
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The impact of Pontryagin and Bohr functors on large-scale properties of locally compact abelian groups (Wikidata QID)
Wikidata QID: Q126567418

The impact of Pontryagin and Bohr functors on large-scale properties of locally compact abelian groups (DOI)
DOI: 10.1016/J.TOPOL.2019.107025

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