A Counterexample to Generalized Accessibility

Abstract

Throughout this paper, G ×T → T will be an action of the group G on the tree T. Unless noted, our actions are simplicial and without inversions. In [BF], we showed that if a) G is finitely presented and b) G ×T → T satisfies certain other conditions reviewed below, then the number of vertices and edges of T/G is bounded in terms of G alone. The main point of this paper is to produce actions G ₀ ×T → T of a fixed finitely generated group which satisfy b) and yet have arbitrarily many vertices. By taking the limit of a sequence of more and more complicated actions, we obtain an interesting action of G _o on an ℝ-tree.