Intermediate geodesic growth in virtually nilpotent groups

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 sub-exponential. 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 provide an example of virtually $3$-step nilpotent group for which $γ_{\mathrm{geod}}(n) \asymp \exp\!\big(n^{3/5}\cdot \log(n)\big)$. This is the first known 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.