Article
Parabolic subgroups acting on the additional length graph
Author/s | Antolín, Yago
Cumplido Cabello, María |
Department | Universidad de Sevilla. Departamento de Álgebra |
Publication Date | 2021-08-18 |
Deposit Date | 2022-07-06 |
Published in |
|
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 ... 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. |
Citation | Antolín, Y. y Cumplido Cabello, M. (2021). Parabolic subgroups acting on the additional length graph. Algebraic & Geometric Topology, 21 (4), 1791-1816. |
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