This note records some observations concerning geodesic growth
functions. If a nilpotent group is not virtually cyclic then it has exponential
geodesic growth with respect to all finite generating sets. On the other hand,
if a finitely generated group G has an element whose normal closure is abelian
and of finite index, then G has a finite generating set with respect to which
the geodesic growth is polynomial (this includes all virtually cyclic groups)
Bridson, M. [et al.]. On groups whose geodesic growth is polynomial. "International journal of algebra and computation", Agost 2012, vol. 22, núm. 5, p. 1-11.