Growth series for Artin groups of dihedral type

Jean Mairesse (1) , Frédéric Mathéus (2)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 LMAM - Laboratoire de Mathématiques et Applications des Mathématiques, EA 3885

We consider the Artin groups of dihedral type I2(k) defined by the presentation Ak = < 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 Ak with respect to the Artin generators (a,b, and inverses) are rational. We provide explicit formulas for the series.

Mots clés: Artin group of dihedral type; spherical growth series; geodesic growth series; automatic group; translation length.; translation length

Domaines: Combinatoire [math.CO]; Théorie des groupes [math.GR]

Jean Mairesse, Frédéric Mathéus. Growth series for Artin groups of dihedral type. International Journal of Algebra and Computation, 2006, 16 (6), pp.1087-1107. ⟨hal-00164813⟩