Conjugacy geodesics and growth in dihedral Artin groups

In this paper we describe conjugacy geodesic representatives in any dihedral Artin group $G(m)$, m≥ 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 , y\ is transcendental. This, together with recent results on Artin groups and contracting elements, implies that all Artin groups of XXL-type have transcendental conjugacy growth series for some generating set. 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 , y\. These imply that the language of all conjugacy geodesics in $G(m)$ with respect to , y\ is regular.