Controlled Connectivity for Semi-Direct Products Acting on Locally Finite Trees

In 2003 Bieri and Geoghegan generalized the Bieri-Neuman-Strebel invariant $\Sigma^1$ by defining $\Sigma^1(\rho)$, $\rho$ an isometric action by a finitely generated group $G$ on a proper CAT(0) space $M$. In this paper, we show how the natural and well-known connection between Bass-Serre theory and covering space theory provides a framework for the calculation of $\Sigma^1(\rho)$ when $\rho$ is a cocompact action by $G = B \rtimes A$, $A$ a finitely generated group, on a locally finite Bass-Serre tree $T$ for $A$. This framework leads to a theorem providing conditions for including an endpoint in, or excluding an endpoint from, $\Sigma^1(\rho)$. When $A$ is a finitely generated free group acting on its Cayley graph, we can restate this theorem from a more algebraic perspective, which leads to some general results on $\Sigma^1$ for such actions.