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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| Date: | 2021 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2021
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1737 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543424038043648 |
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| author | Protasov, I. Protasova, K. |
| author_facet | Protasov, I. Protasova, K. |
| author_sort | Protasov, I. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2021-11-09T03:53:16Z |
| 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. |
| first_indexed | 2026-02-08T07:59:21Z |
| format | Article |
| id | admjournalluguniveduua-article-1737 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:59:21Z |
| publishDate | 2021 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-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 |
| spellingShingle | coarse structure defined by conjugations cellularity FC-group ultrafilter 20E45 54D80 Protasov, I. Protasova, K. Coarse structures on groups defined by conjugations |
| title | 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_short | Coarse structures on groups defined by conjugations |
| title_sort | coarse structures on groups defined by conjugations |
| topic | coarse structure defined by conjugations cellularity FC-group ultrafilter 20E45 54D80 |
| topic_facet | coarse structure defined by conjugations cellularity FC-group ultrafilter 20E45 54D80 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1737 |
| work_keys_str_mv | AT protasovi coarsestructuresongroupsdefinedbyconjugations AT protasovak coarsestructuresongroupsdefinedbyconjugations |