On fibers and accessibility of groups acting on trees with inversions

• Date: 2015
• Main Author: Mahmood, Rasheed Mahmood Saleh
• Format: Article
• Language: English
• Published: Lugansk National Taras Shevchenko University 2015
• Subjects: Ends of groups; groups acting on trees; accessible groups; 20E06; 20E086; 20F05
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/66
• Journal Title: Algebra and Discrete Mathematics

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.