A virtually 2-step nilpotent group with polynomial geodesic growth

TL;DR

This paper constructs an example of a virtually 2-step nilpotent group exhibiting polynomial geodesic growth, expanding known cases beyond virtually abelian groups and illustrating the diversity of group growth behaviors.

Contribution

It provides the first known example of a virtually 2-step nilpotent group with polynomial geodesic growth, beyond virtually abelian groups.

Findings

Demonstrates a specific virtually 2-step nilpotent group with polynomial geodesic growth

Expands the class of groups known to have polynomial geodesic growth

Highlights the diversity of growth behaviors in virtually nilpotent groups

Abstract

A direct consequence of Gromov's theorem is that if a group has polynomial geodesic growth with respect to some finite generating set then it is virtually nilpotent. However, until now the only examples known were virtually abelian. In this note we furnish an example of a virtually 2-step nilpotent group having polynomial geodesic growth with respect to a certain finite generating set.