Geodesic
On fibers and accessibility of groups acting on trees with inversions
Algebra and discrete mathematics, Tome 19 (2015) no. 2, pp. 229-242

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Throughout this paper the actions of groups on graphs with inversions are allowed. 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 there exists a tree on which $G$ acts such that each edge group is finite, no vertex is 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 if for 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 contains no 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 fiber tree 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 each edge $e$ of $X$ is finite and contains no inverter elements of $G_{t(e)}$, the vertex $G_{v}$ group of each vertex $v$ of $X$ is accessible, and the quotient graph $G\diagup X$ for the action of $G$ on $X$ is finite, then $G$ is an accessible group.
Keywords: ends of groups, groups acting on trees, accessible groups. 
Rasheed Mahmood Saleh Mahmood. On fibers and accessibility of groups acting on trees with inversions. Algebra and discrete mathematics, Tome 19 (2015) no. 2, pp. 229-242. http://geodesic.mathdoc.fr/item/ADM_2015_19_2_a6/
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