Connectivity Properties for Actions on Locally Finite Trees

Given an action by a finitely generated group G on a locally finite tree T, we view points of the visual boundary \partialT as directions in T and use {\rho} to lift this sense of direction to G. For each point E \in \partialT, this allows us to ask if G is (n - 1)-connected "in the direction of E". The invariant {\Sigma}^n({\rho}) \subseteq \partialT then records the set of directions in which G is (n-1)-connected. In this paper, we introduce a family of actions for which {\Sigma}^1({\rho}) can be calculated through analysis of certain quotient maps between trees. We show that for actions of this sort, under reasonable hypotheses, {\Sigma}1({\rho}) 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 {\Sigma}^n({\rho}) for any n.