On the fundamental domains of the fibers of groups acting on trees with inversions

Abstract

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. If G is a group acting on a tree X with inversions, T and Y are two subtrees of X such that T⊆Y, we call T a tree of representatives for the action of G on X if T contains exactly one vertex from each vertex orbit, and call Y a transversal for the action of G on X if each edge of Y has at least one end in T, and Y satisfies the conditions that Y contains exactly one edge y from edge orbit if y and its inverse y are not in the same edge orbit and exactly one pair of an edge from each edge orbit if x and x are in the same edge orbit . In symbols, we say that (T; Y) is a fundamental domain for the action of G on X. In this paper we show that if G is a group acting on a tree X with a given fundamental domain (T;Y), and if for each vertex v of X, the stabilizer v G of the vertex v acts on a tree v X with a given fundamental domain ) ; ( v v Y T for the action of v G on v X such that the edge stabilizer e G of each edge e of X, is finite and contains no inverter element of the stabilizer ) (e o G of the terminal o(e) of e, then we obtain a new tree X ~ called a fiber tree such that G acts on X~ with a fundamental domain ) ~ ; ~ ( Y T for the action of G on X~ . As an application, we show that the fundamental group ) ~ ( X G π of the quotient graph X G ~ of the action of G on X~ is isomorphic to the free product of the fundamental groups ) ( X G π and ) ( v v X G π of the quotient graphs X G and v v X G of the actions of G on X and v G on ) ( , T V v X v ∈ respectively. Mathematics Subject Classification: 20E08, 20E06, 20F05 400 R. M. S. Mahmood