Given an action
GT 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 whether G is
(n− 1)-connected “in the direction of E.” Then the invariant Σn(ρ) ⊆ ∂T records the
set of directions in which G is (n − 1)-connected. We introduce a family of
actions for which Σ1(ρ) can be calculated through analysis of certain quotient
maps between trees. We show that for actions of this sort, under reasonable
hypotheses, Σ1(ρ) consists of no more than a single point. By strengthening the
hypotheses, we characterize precisely when a given end point lies in Σn(ρ) for any
n.
