This article is cited in 1 paper

RESEARCH ARTICLE

Coarse structures on groups defined by conjugations

I. Protasov, K. Protasova

Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Academic Glushkov pr. 4d, 03680 Kyiv, Ukraine

Abstract: For a group $G$, we denote by $\stackrel{\leftrightarrow}{G}$ the coarse space on $G$ endowed with the coarse structure with the base $\{\{(x,y)\in G\times G\colon y\in x^F \} \colon F \in [G]^{<\omega} \}$, $x^F = \{z^{-1} xz\colon z\in F \}$. Our goal is to explore interplays between algebraic properties of $G$ and asymptotic properties of $\stackrel{\leftrightarrow}{G}$. In particular, we show that $\operatorname{asdim}\stackrel{\leftrightarrow}{G} = 0$ if and only if $G / Z_G$ is locally finite, $Z_G$ is the center of $G$. For an infinite group $G$, the coarse space of subgroups of $G$ is discrete if and only if $G$ is a Dedekind group.

Keywords: coarse structure defined by conjugations, cellularity, FC-group, ultrafilter.

MSC: 20E45, 54D80

Received: 12.12.2020
Revised: 21.03.2021

Language: English

DOI: 10.12958/adm1737