Parabolic subgroups acting on the additional length graph
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 construc...
| Autores: | , |
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2021 |
| País: | España |
| Recursos: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/135026 |
| Acesso em linha: | https://hdl.handle.net/11441/135026 https://doi.org/10.2140/agt.2021.21.1791 |
| Access Level: | acceso abierto |
| Palavra-chave: | braid groups Artin groups Garside groups parabolic subgroups acylindrically hyperbolic groups growth of groups relative growth |
| Resumo: | 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. |
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