Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles

TL;DR

This paper investigates the growth properties and almost convexity of higher-dimensional torus bundle groups, showing conditions under which they have rational growth series or lack almost convexity.

Contribution

It establishes that certain torus bundle groups have rational growth series when associated matrices have distinct eigenvalues off the unit circle, and are not almost convex otherwise.

Findings

Groups with matrices having eigenvalues off the unit circle have rational growth series.

Such groups are not almost convex for any generating set.

The results connect eigenvalue properties to geometric group theory features.

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.