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 Gv of v acts on a tree Xv, the edge group Ge of each edge e of X is finite and contains no inverter elements of the vertex group Gt(e) of the terminal t(e) of e, then we obtain a new tree denoted X̃ and is called a fiber tree such that G acts on X̃. As an application, we show that if G is a group acting on a tree X such that the edge group Ge for each edge e of X is finite and contains no inverter elements of Gt(e), the vertex Gv group of each vertex v of X is accessible, and the quotient graph G X for the action of G on X is finite, then G is an accessible group. The author would like to thank the referee for his(her) help and suggestions to improve the first draft of this paper. 2000 MSC: 20E06, 20E086, 20F05.