Geodesic growth in virtually abelian groups

TL;DR

This paper investigates the geodesic growth in virtually abelian groups, establishing that it is either polynomial or exponential, with the growth series being holonomic and rational in the polynomial case, and characterizing the geodesic language as blind multicounter.

Contribution

It proves the dichotomy of geodesic growth types and characterizes the geodesic language structure in virtually abelian groups.

Findings

Geodesic growth is either polynomial or exponential.

Growth series is holonomic and rational in polynomial case.

Geodesic language is blind multicounter.

Abstract

We show that the geodesic growth function of any finitely generated virtually abelian group is either polynomial or exponential; and that the geodesic growth series is holonomic, and rational in the polynomial growth case. In addition, we show that the language of geodesics is blind multicounter.