On fibers and accessibility of groups acting on trees with inversions
Throughout this paper the actions of groups on graphs with inversions areallowed. An element g of a group \(G\) is called inverter if there exists a tree\(X\) where \(G\) acts such that \(g\) transfers an edge of \(X\) into its inverse.\(A\) group \(G\) is called accessible if \(G\) is finitely gene...
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| Дата: | 2015 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2015
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/66 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-662015-09-28T11:22:08Z On fibers and accessibility of groups acting on trees with inversions Mahmood, Rasheed Mahmood Saleh Ends of groups, groups acting on trees, accessible groups 20E06, 20E086, 20F05 Throughout this paper the actions of groups on graphs with inversions areallowed. An element g of a group \(G\) is called inverter if there exists a tree\(X\) where \(G\) acts such that \(g\) transfers an edge of \(X\) into its inverse.\(A\) group \(G\) is called accessible if \(G\) is finitely generated and thereexists a tree on which \(G\) acts such that each edge group is finite, no vertexis stabilized by $G$, and each vertex group has at most one end.In this paper we show that if \(G\) is a group acting on a tree \(X\) such that iffor each vertex \(v\) of \(X\), the vertex group \(G_{v}\) of \(v\) acts on a tree\(X_{v}\), the edge group \(G_{e}\) of each edge e of \(X\) is finite and containsno inverter elements of the vertex group \(G_{t(e)}\) of the terminal \(t(e)\) of$e$, then we obtain a new tree denoted \(\widetilde{X}\) and is called a fibertree such that \(G\) acts on \(\widetilde{X}\). As an application, we show that if$G$ is a group acting on a tree \(X\) such that the edge group \(G_{e}\) for eachedge \(e\) of \(X\) is finite and contains no inverter elements of \(G_{t(e)}\), thevertex \(G_{v}\) group of each vertex \(v\) of \(X\) is accessible, and the quotientgraph \(G\diagup X\) for the action of \(G\) on \(X\) is finite, then \(G\) is anaccessible group. Lugansk National Taras Shevchenko University 2015-09-28 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/66 Algebra and Discrete Mathematics; Vol 19, No 2 (2015) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/66/16 Copyright (c) 2015 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2015-09-28T11:22:08Z |
| collection |
OJS |
| language |
English |
| topic |
Ends of groups groups acting on trees accessible groups 20E06 20E086 20F05 |
| spellingShingle |
Ends of groups groups acting on trees accessible groups 20E06 20E086 20F05 Mahmood, Rasheed Mahmood Saleh On fibers and accessibility of groups acting on trees with inversions |
| topic_facet |
Ends of groups groups acting on trees accessible groups 20E06 20E086 20F05 |
| format |
Article |
| author |
Mahmood, Rasheed Mahmood Saleh |
| author_facet |
Mahmood, Rasheed Mahmood Saleh |
| author_sort |
Mahmood, Rasheed Mahmood Saleh |
| title |
On fibers and accessibility of groups acting on trees with inversions |
| title_short |
On fibers and accessibility of groups acting on trees with inversions |
| title_full |
On fibers and accessibility of groups acting on trees with inversions |
| title_fullStr |
On fibers and accessibility of groups acting on trees with inversions |
| title_full_unstemmed |
On fibers and accessibility of groups acting on trees with inversions |
| title_sort |
on fibers and accessibility of groups acting on trees with inversions |
| description |
Throughout this paper the actions of groups on graphs with inversions areallowed. An element g of a group \(G\) is called inverter if there exists a tree\(X\) where \(G\) acts such that \(g\) transfers an edge of \(X\) into its inverse.\(A\) group \(G\) is called accessible if \(G\) is finitely generated and thereexists a tree on which \(G\) acts such that each edge group is finite, no vertexis stabilized by $G$, and each vertex group has at most one end.In this paper we show that if \(G\) is a group acting on a tree \(X\) such that iffor each vertex \(v\) of \(X\), the vertex group \(G_{v}\) of \(v\) acts on a tree\(X_{v}\), the edge group \(G_{e}\) of each edge e of \(X\) is finite and containsno inverter elements of the vertex group \(G_{t(e)}\) of the terminal \(t(e)\) of$e$, then we obtain a new tree denoted \(\widetilde{X}\) and is called a fibertree such that \(G\) acts on \(\widetilde{X}\). As an application, we show that if$G$ is a group acting on a tree \(X\) such that the edge group \(G_{e}\) for eachedge \(e\) of \(X\) is finite and contains no inverter elements of \(G_{t(e)}\), thevertex \(G_{v}\) group of each vertex \(v\) of \(X\) is accessible, and the quotientgraph \(G\diagup X\) for the action of \(G\) on \(X\) is finite, then \(G\) is anaccessible group. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2015 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/66 |
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2025-07-17T10:30:23Z |
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