Twisted conjugacy growth series of virtually abelian groups

TL;DR

This paper introduces the concept of twisted conjugacy growth series in finitely generated groups and proves that for virtually abelian groups, this series is always a computable rational function, extending to relative growth series of conjugacy classes.

Contribution

It is the first to define and analyze twisted conjugacy growth series, showing they are rational for virtually abelian groups, providing explicit computability.

Findings

Twisted conjugacy growth series are rational functions for virtually abelian groups.

The series can be explicitly computed.

Results extend to relative growth series of conjugacy classes.

Abstract

We initiate the study of the \emph{twisted conjugacy growth series} of a finitely generated group, the formal power series associated to the twisted conjugacy growth function. Our main result is that, for a virtually abelian group, this series is always an explicitly computable $\mathbb{N}$-rational function. As a corollary, we obtain a similar result for the relative growth series of a twisted conjugacy class in a virtually abelian group.