Antolín, YagoCumplido Cabello, María2022-07-062022-07-062021-08-18Antolín, Y. y Cumplido Cabello, M. (2021). Parabolic subgroups acting on the additional length graph. Algebraic & Geometric Topology, 21 (4), 1791-1816.1472-2739https://hdl.handle.net/11441/135026Let 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.application/pdf19 p.engAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/braid groupsArtin groupsGarside groupsparabolic subgroupsacylindrically hyperbolic groupsgrowth of groupsrelative growthinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccesshttps://doi.org/10.2140/agt.2021.21.1791