Extra-large type Artin groups are hierarchically hyperbolic

TL;DR

This paper proves that Artin groups of extra-large and hyperbolic types are hierarchically hyperbolic, leading to finite asymptotic dimension and exponential growth, using a new combinatorial approach and complex constructions.

Contribution

It introduces a novel combinatorial method to establish hierarchical hyperbolicity for large classes of Artin groups, expanding understanding of their geometric properties.

Findings

Artin groups of extra-large and hyperbolic type are hierarchically hyperbolic

These groups have finite asymptotic dimension

They exhibit uniform exponential growth

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-Gonz\'alez-Meneses-Wiest.