### Coarse structures on groups defined by conjugations

#### 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.

#### Keywords

coarse structure defined by conjugations, cellularity, FC-group, ultrafilter

#### Full Text:

PDFDOI: http://dx.doi.org/10.12958/adm1737

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