Parabolic subgroups acting on the additional length graph

Abstract

Let $A\ne A_1,A_2,I_{2m}$ 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 ${\mathcal{𝒞}}_{AL}\left(A\right)$, a hyperbolic, infinite diameter graph associated to $A$ constructed by Calvez and Wiest to show that $A∕Z\left(A\right)$ is acylindrically hyperbolic. We use these results to find an element $g\in A$ such that $⟨P,g⟩\cong P\ast ⟨g⟩$ 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 ${\left\{\omega \left(A_n,\mathcal{𝒮}\right)\right\}}_{n\in \mathbb{N}}$ of exponential growth rates of braid groups, with respect to the Garside generating set, goes to infinity.