Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles

Abstract

Given a matrix $A\in SL(N,\mathbb{Z})$, form the semidirect product $G={\mathbb{Z}}^N{⋊}_A\mathbb{Z}$ where the $\mathbb{Z}$ factor acts on ${\mathbb{Z}}^N$ 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 $H\le G$ 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.

31 pages. Added a reference and a remark concerning previous work on the result in Theorem 1.2