Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles

Abstract

Given a matrix $A\in SL(N,\Z)$, form the semidirect product $G=\Z^N\rtimes_A \Z$ where the $\Z$ factor acts on $\Z^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this paper we prove that if $A$ has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup $H\leq G$ possessing rational growth series for some generating set. In contrast, we show that if $A$ has at least one eigenvalue not lying on the unit circle, then $G$ is not almost convex for any generating set.