Conjugacy geodesics and growth in dihedral Artin groups
Laura Ciobanu, Gemma Crowe
TL;DR
The paper computes conjugacy geodesic representatives for dihedral Artin groups and uses these to derive precise conjugacy-growth asymptotics, showing transcendence of the conjugacy-growth series in the free-product generating set. It introduces and exploits a constant permutation conjugator length and the FFTP to prove regularity of the conjugacy-geodesic language, with the result that ConjGeo$(G(m),\{x,y\})$ is regular. Across both odd and even cases, the authors relate conjugacy growth to standard growth, proving that the conjugacy growth rate equals the standard growth rate, and establish a unified framework that handles non-uniqueness of geodesics via split cyclic permutations. The methods combine detailed geodesic classifications, central-element gymnastics, and analytic combinatorics to obtain transcendence results and regularity statements that extend understanding of conjugacy growth in non-virtually-abelian groups. The findings also yield implications for related groups like BS$(p,p)$ and braid groups, illustrating broad applicability of the conjugacy-geodesic approach.
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.