GROWTH SERIES FOR ARTIN GROUPS OF DIHEDRAL TYPE

Abstract

We consider the Artin groups of dihedral type I 2 (k) defined by the presentation A k = 〈a,b | prod (a,b;k) = prod (b,a;k)〉 where prod (s,t;k) = ststs …, with k terms in the product on the right-hand side. We prove that the spherical growth series and the geodesic growth series of A k with respect to the Artin generators {a,b,a -1 , b -1 } are rational. We provide explicit formulas for the series.