Rational Growth in Virtually Abelian Groups

TL;DR

This paper proves that various growth functions related to subgroups, cosets, and conjugacy classes in finitely generated virtually abelian groups are rational, regardless of the generating set used.

Contribution

It establishes the rationality of subgroup, coset, and conjugacy class growth functions in virtually abelian groups, independent of generating set choice.

Findings

Subgroup growth in virtually abelian groups is rational.

Coset growth functions are rational in these groups.

Conjugacy class growth functions are rational regardless of generating set.

Abstract

We show that any subgroup of a finitely generated virtually abelian group $G$ grows rationally relative to $G$, that the set of right cosets of any subgroup of $G$ grows rationally, and that the set of conjugacy classes of $G$ grows rationally. These results hold regardless of the choice of finite weighted generating set for $G$.