Connectivity properties for actions on locally finite trees

Abstract

Given an action G ρ y T by a finitely generated group on a locally finite tree, we view points of the visual boundary ∂T as directions in T and use ρ to lift this sense of direction to G. For each point E ∈ ∂T , this allows us to ask if G is (n− 1)-connected “in the direction of E”. The invariant Σ(ρ) ⊆ ∂T then records the set of directions in which G is (n−1)-connected. In this paper, we introduce a family of actions for which Σ(ρ) can be calculated through analysis of certain quotient maps between trees. We show that for actions of this sort, under reasonable hypotheses, Σ(ρ) consists of no more than a single point. By strengthening the hypotheses, we are able to characterize precisely when a given end point lies in Σ(ρ) for any n.