Extra-large type Artin groups are hierarchically hyperbolic

Abstract

We show that Artin groups of extra-large type, and more generally Artin groups of large and hyperbolic type, are hierarchically hyperbolic. This implies in particular that these groups have finite asymptotic dimension and uniform exponential growth.We prove these results by using a combinatorial approach to hierarchical hyperbolicity, via the action of these groups on a new complex that is quasi-isometric both to the coned-off Deligne complex introduced by Martin-Przytycki and to a generalisation due to Morris-Wright of the graph of irreducible parabolic subgroups of finite type introduced by Cumplido-Gebhardt-González-Meneses-Wiest.Contents 2.1. Artin groups 2.2. Structure of dihedral Artin groups 2.3. The modified Deligne complex 2.4. Standard trees and the coned-off Deligne complex 2.5. Links of vertices 3. The commutation graph of an Artin group 3.1. The commutation graph 3.2. The graph of proper irreducible parabolic subgroups of finite type 3.3. Adjacency in the commutation graph and intersection of neighbourhoods of cosets 4. The blown-up commutation graph 4.1. Blow-up data 4.2. The blown-up commutation graph 4.3. Simplices of the blow-up and their links 4.4. From maximal simplices to elements of A Γ 5. The augmented complex 5.1. Construction of the augmented complex 5.2. Fullness of links 5.3. Maps between augmented complexes 6. The geometry of the augmented complex

Asymptotic dimension of Artin groups and a new upper bound for Coxeter groups

“…[8] Do all Out(F n ) groups have finite asymptotic dimension, when n > 2? [8] Recently, it was proved in [14] that Artin groups being both of large and of hyperbolic type are hierarchically hyperbolic. As a consequence, these Artin groups have finite asymptotic dimension.…”

Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms

“…This explains the involvement of quasimorphisms in our proof. The idea of using quasimorphisms in building HHG structures originated in this project, but has already found additional applications to Artin groups [HMS21] and extensions of subgroups of mapping class groups [DDLS20].…”

“…This technique of building combinatorial HHSs by "blowing up the vertex groups" in some naturally-occurring hyperbolic graph is quite flexible, and has analogues in a number of other contexts. For example, it is applied in the context of certain Artin groups in [HMS21], extensions of lattice Veech groups in [DDLS20], and extensions of multicurve stabilisers in [Rus21]. In [BHMS20], it is explained how to build combinatorial HHSs for right-angled Artin groups and mapping class groups by respectively blowing up the Kim-Koberda extension graph [KK13] and the curve graph.…”