Intermediate geodesic growth in virtually nilpotent groups

Abstract

We give a criterion on pairs $(G,S)$ – where $G$ is a virtually $s$-step nilpotent group and $S$ is a finite generating set – saying whether the geodesic growth is exponential or strictly subexponential. Whenever $s=1,2$, this goes further, and we prove the geodesic growth is either exponential or polynomial. For $s\ge 3$, however, intermediate growth is possible. We exhibit a pair $(G,S)$ for which ${\gamma }_{\mathrm{geod}}(n)\asymp \exp (n^{3/5}\cdot \log (n))$, where $G$ contains a $3$-step nilpotent group – the Engel group – as a finite-index subgroup. This is the first example of group with intermediate geodesic growth. Along the way, we prove results on the geometry of virtually nilpotent groups, including asymptotics with error terms for their volume growth, and disprove a conjecture by Breuillard and Le Donne.