#### Volume 21, issue 4 (2021)

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Parabolic subgroups acting on the additional length graph

### Yago Antolín and María Cumplido

Algebraic & Geometric Topology 21 (2021) 1791–1816
##### 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 ${\mathsc{𝒞}}_{AL}\left(A\right)$, 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 ${\left\{\omega \left({A}_{n},\mathsc{𝒮}\right)\right\}}_{n\in ℕ}$ of exponential growth rates of braid groups, with respect to the Garside generating set, goes to infinity.

##### Keywords
braid groups, Artin groups, Garside groups, parabolic subgroups, acylindrically hyperbolic groups, growth of groups, relative growth
##### Mathematical Subject Classification 2010
Primary: 20F36, 20F65