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We give a criterion on pairs $(G,S)$ - where $G$ is a virtually $s$-stepnilpotent group and $S$ is a finite generating set - saying whether thegeodesic growth is exponential or strictly sub-exponential. Whenever $s=1,2$,this goes further and we prove the geodesic growth is either exponential orpolynomial. For $s\ge 3$ however, intermediate growth is possible. We providean example of virtually $3$-step nilpotent group for which$\gamma_{\mathrm{geod}}(n) \asymp \exp\!\big(n^{3/5}\cdot \log(n)\big)$. Thisis the first known example of group with intermediate geodesic growth. Alongthe way, we prove results on the geometry of virtually nilpotent groups,including asymptotics with error terms for their volume growth.

Intermediate geodesic growth in virtually nilpotent groups