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

Abstract

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 $G=B⋊A$, $A$ 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.