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 {x, 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 {x, y}. These imply that the language of all conjugacy geodesics in G(m) with respect to {x, y} is regular.