Conjugacy growth series and languages in groups

TL;DR

This paper introduces new concepts related to conjugacy in finitely generated groups, analyzing how various group constructions affect the rationality and regularity of associated growth series and languages.

Contribution

It defines the geodesic conjugacy language and growth series, and studies their properties under different group constructions, highlighting preservation of regularity and rationality.

Findings

Regularity of geodesic conjugacy language preserved by graph products.

Rationality of geodesic conjugacy growth series preserved by direct and free products.

New frameworks for analyzing conjugacy growth in groups.

Abstract

In this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series, as well as on regularity of the geodesic conjugacy language and spherical conjugacy language. In particular, we show that regularity of the geodesic conjugacy language is preserved by the graph product construction, and rationality of the geodesic conjugacy growth series is preserved by both direct and free products.