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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| Мова: | English |
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Lugansk National Taras Shevchenko University
2021
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-17372021-11-09T03:53:16Z Coarse structures on groups defined by conjugations Protasov, I. Protasova, K. coarse structure defined by conjugations, cellularity, FC-group, ultrafilter 20E45, 54D80 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. Lugansk National Taras Shevchenko University 2021-11-09 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1737 10.12958/adm1737 Algebra and Discrete Mathematics; Vol 32, No 1 (2021) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1737/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1737/803 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1737/845 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1737/846 Copyright (c) 2021 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2021-11-09T03:53:16Z |
| collection |
OJS |
| language |
English |
| topic |
coarse structure defined by conjugations cellularity FC-group ultrafilter 20E45 54D80 |
| spellingShingle |
coarse structure defined by conjugations cellularity FC-group ultrafilter 20E45 54D80 Protasov, I. Protasova, K. Coarse structures on groups defined by conjugations |
| topic_facet |
coarse structure defined by conjugations cellularity FC-group ultrafilter 20E45 54D80 |
| format |
Article |
| author |
Protasov, I. Protasova, K. |
| author_facet |
Protasov, I. Protasova, K. |
| author_sort |
Protasov, I. |
| title |
Coarse structures on groups defined by conjugations |
| title_short |
Coarse structures on groups defined by conjugations |
| title_full |
Coarse structures on groups defined by conjugations |
| title_fullStr |
Coarse structures on groups defined by conjugations |
| title_full_unstemmed |
Coarse structures on groups defined by conjugations |
| title_sort |
coarse structures on groups defined by conjugations |
| description |
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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2021 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1737 |
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AT protasovi coarsestructuresongroupsdefinedbyconjugations AT protasovak coarsestructuresongroupsdefinedbyconjugations |
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2025-07-17T10:34:20Z |
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2025-07-17T10:34:20Z |
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