Coarse structures on groups defined by conjugations

Igor Protasov; Ksenia Protasova

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arXiv · 2008.11011

Coarse structures on groups defined by conjugations

Igor Protasov·Ksenia Protasova

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Abstract

For a group $G$, we denote by $\stackrel{\leftrightarrow}{G}$ the coarse space on $G$ endowed with the coarse structure with the base $\{\{ (x,y)\in G\times G: y\in x^F \} : F \in [G]^{<\omega} \}$, $x^F = \{z^{-1} xz : z\in F \}$. Our goal is to explore interplays between algebraic properties of $G$ and asymptotic properties of $\stackrel{\leftrightarrow}{G}$. In particular, we show that $asdim \ \stackrel{\leftrightarrow}{G} = 0$ 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.

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Igor Protasov, Ksenia Protasova. 2020-08-03. Coarse structures on groups defined by conjugations. https://arxiv.org/abs/2008.11011

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