The impact of Pontryagin and Bohr functors on large-scale properties of locally compact abelian groups
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.
scientific article; zbMATH DE number 7187672
scientific article