Connectivity Properties for Actions on Locally Finite Trees

TL;DR

This paper investigates the connectivity properties of group actions on locally finite trees, introducing an invariant that captures directions of connectivity and providing methods to compute and characterize it.

Contribution

It introduces a new invariant {oldsymbol{ ext{ extSigma}}}^n({ ho}) for actions on trees and develops techniques to compute and characterize this invariant in specific cases.

Findings

{oldsymbol{ extSigma}}^1({ ho}) can be computed via quotient maps between trees.

Under certain conditions, {oldsymbol{ extSigma}}^1({ ho}) contains at most one point.

Strengthened hypotheses allow precise characterization of directions in {oldsymbol{ extSigma}}^n({ ho}).

Abstract

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.