Controlled connectivity for semidirect products acting on locally finite trees

Keith Jones

Vol. 268 (2014), No. 1, 79–94

DOI: 10.2140/pjm.2014.268.79

Abstract

In 2003 Bieri and Geoghegan generalized the Bieri–Neumann–Strebel invariant ${\Sigma }^1$ by defining ${\Sigma }^1\left(\rho \right)$, $\rho$ 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 ${\Sigma }^1\left(\rho \right)$ 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, ${\Sigma }^1\left(\rho \right)$. 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.

Keywords

controlled connectivity, BNS, sigma invariants, tree actions, semidirect products

Mathematical Subject Classification 2010

Primary: 20E08, 20F65, 57M07

Secondary: 05C05, 05C25

Milestones

Received: 20 September 2012

Revised: 29 November 2013

Accepted: 9 December 2013

Published: 21 May 2014