Parabolic subgroups acting on the additional length graph

Abstract

Let A ≠ A 1 , A 2 , I 2 m be an irreducible Artin–Tits group of spherical type. We show that the periodic elements of A and the elements preserving some parabolic subgroup of A act elliptically on the additional length graph C AL ( A ) , a hyperbolic, infinite diameter graph associated to A constructed by Calvez and Wiest to show that A ∕ Z ( A ) is acylindrically hyperbolic. We use these results to find an element g ∈ A such that ⟨ P , g ⟩ ≅ P ∗ ⟨ g ⟩ for every proper standard parabolic subgroup P of A . The length of g is uniformly bounded with respect to the Garside generators, independently of A . This allows us to show that, in contrast with the Artin generators case, the sequence { ω ( A n , S ) } n ∈ N of exponential growth rates of braid groups, with respect to the Garside generating set, goes to infinity.