Conjugacy geodesics and growth in dihedral Artin groups

TL;DR

This paper characterizes conjugacy geodesics in dihedral Artin groups, computes their conjugacy growth asymptotics, and proves the conjugacy geodesic language is regular, revealing new algebraic and combinatorial properties.

Contribution

It provides explicit descriptions of conjugacy geodesics, calculates conjugacy growth series, and establishes regularity of the conjugacy geodesic language in dihedral Artin groups.

Findings

Conjugacy geodesic representatives are explicitly described.

Conjugacy growth series is transcendental.

The conjugacy geodesic language is regular.

Abstract

In this paper we describe conjugacy geodesic representatives in any dihedral Artin group $G(m)$, $m\geq 3$, which we then use to calculate asymptotics for the conjugacy growth of $G(m)$, and show that the conjugacy growth series of $G(m)$ with respect to the `free product' generating set $\{x, y\}$ is transcendental. We prove two additional properties of $G(m)$ that connect to conjugacy, namely that the permutation conjugator length function is constant, and that the falsification by fellow traveler property (FFTP) holds with respect to $\{x, y\}$. These imply that the language of all conjugacy geodesics in $G(m)$ with respect to $\{x, y\}$ is regular.