On groups whose geodesic growth is polynomial

Martin Bridson, Jose Burillo, Murray Elder, Zoran Sunic

Published 2010-09-26, updated 2012-03-06Version 3

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).