Conjugacy Class Growth in Virtually Abelian Groups

TL;DR

This paper investigates the growth of conjugacy classes in finitely generated virtually abelian groups, proving polynomial asymptotic behavior, and relates conjugacy class growth in affine Coxeter groups to reflection length.

Contribution

It establishes the polynomial nature of conjugacy class growth in virtually abelian groups and links conjugacy class growth degree to reflection length in affine Coxeter groups.

Findings

Conjugacy class growth in virtually abelian groups is asymptotically polynomial.

In affine Coxeter groups, growth degree correlates with reflection length.

Provides a unified view of conjugacy class growth across different group classes.

Abstract

We study the conjugacy class growth function in finitely generated virtually abelian groups. That is, the number of elements in the ball of radius $n$ in the Cayley graph which intersect a fixed conjugacy class. In the class of virtually abelian groups, we prove that this function is always asymptotically equivalent to a polynomial. Furthermore, we show that in any affine Coxeter group, the degree of polynomial growth of a conjugacy class is equivalent to the reflection length of any element of that class.