Which finitely generated Abelian groups admit equal growth functions?

TL;DR

This paper characterizes when finitely generated Abelian groups have equal growth functions, revealing that rank and torsion parity determine equality for symmetric generating sets, while rank alone suffices for monoid generating sets.

Contribution

It provides a complete classification of finitely generated Abelian groups based on their growth functions with respect to different generating sets.

Findings

Equal growth functions with symmetric sets require same rank and torsion parity.

Equal growth functions with monoid sets require only same rank.

Torsion part size is determined by all growth functions of the group.

Abstract

We show that finitely generated Abelian groups admit equal growth functions with respect to symmetric generating sets if and only if they have the same rank and the torsion parts have the same parity. In contrast, finitely generated Abelian groups admit equal growth functions with respect to monoid generating sets if and only if they have same rank. Moreover, we show that the size of the torsion part is in fact determined by the set of all growth functions of a finitely generated Abelian group.