On fibers and accessibility of groups acting on trees with inversions

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.