Coarse structures on groups defined by conjugations
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...
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| Дата: | 2021 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2021
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| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1737 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| Резюме: | 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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