Automatic structures and growth functions for finitely generated abelian groups

TL;DR

This paper studies the growth functions of finitely generated abelian groups, showing they are rational and explicitly determining their form based on the group's rank, extending prior results on automatic structures.

Contribution

It provides an explicit computation of growth functions for abelian groups, linking the denominator of the rational function to the group's rank, expanding understanding of automatic structures.

Findings

Growth functions for abelian groups are rational functions.

The denominator of the growth function is determined by the group's rank.

The results extend previous work on automatic structures in groups.

Abstract

In this paper, we consider the formal power series whose n-th coefficient is the number of copies of a given finite graph in the ball of radius n centred at the identity element in the Cayley graph of a finitely generated group and call it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the growth function is a rational function if the group has a geodesic automatic structure. We compute the growth function in the case where the group is abelian and see that the denominator of the rational function is determined from the rank of the group.