Parabolic subgroups acting on the additional length graph

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Antolín Pichel, Yago and Cumplido, María (2021) Parabolic subgroups acting on the additional length graph. Algebraic & Geometric Topology, 21 (4). pp. 1791-1816. ISSN 1472-2747

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Official URL: https://doi.org/10.2140/agt.2021.21.1791




Abstract

Let A ≠ A1;A2;I2m 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 CAL(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(An,S)(with n ∈ N) of exponential growth rates of braid groups, with respect to the Garside generating set, goes to infinity.


Item Type:Article
Uncontrolled Keywords:Braid groups; Artin groups; Geometric group theory
Subjects:Sciences > Mathematics > Algebra
Sciences > Mathematics > Group Theory
ID Code:69511
Deposited On:26 Jan 2022 16:48
Last Modified:10 Feb 2022 16:51

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