On groups whose geodesic growth is polynomial

TL;DR

This paper investigates the geodesic growth of groups, showing that non-virtually cyclic nilpotent groups have exponential growth, while certain groups with abelian normal closures can have polynomial geodesic growth.

Contribution

It identifies conditions under which groups exhibit polynomial versus exponential geodesic growth, extending understanding of growth behaviors in group theory.

Findings

Nilpotent non-virtually cyclic groups have exponential geodesic growth

Groups with an element whose normal closure is abelian and finite index can have polynomial geodesic growth

Virtually cyclic groups can have polynomial geodesic growth with suitable generating sets

Abstract

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).