Algebra and Discrete Mathematics On fibers and accessibility of groups acting on trees with inversions

Rasheed Mahmood, Saleh Mahmood
2015 unpublished
A b s t r ac t. 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
more » ... 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 X and is called a fiber tree such that G acts on 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 GX 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.
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