Acylindrical hyperbolicity for Artin groups of dimension 2

Abstract

In this paper, we show that every irreducible 2-dimensional Artin group $$A_{\Gamma }$$ A Γ of rank at least 3 is acylindrically hyperbolic. We do this by studying the action of $$A_\Gamma $$ A Γ on its modified Deligne complex. Along the way, we prove results of independent interests on the geometry of links of this complex.

Extra-large type Artin groups are hierarchically hyperbolic

“…This question has been answered positively for most standard classes of Artin groups, such as right-angled Artin groups [19], Artin groups of finite type [17], Artin groups of Euclidean type [16], Artin groups whose underlying presentation graph is not a join [22], and two-dimensional Artin groups [55].…”

“…The graph of orbits Orbit a,b G is defined as the first barycentric subdivision of the graph obtained in this way. The action of A ab on Lk D (v ab ) has been studied by Vaskou in [55]. In particular, the following result, which is a geometric counterpart of Lemma 2.6 will be useful in Sect.…”

Parabolic subgroups of large-type Artin groups

“…Indeed, several complexes have been associated to Artin groups using the combinatorics of parabolic subgroups. For instance, Deligne complexes and their variants are built out of (cosets of) standard parabolic subgroups of spherical type [6], and have been used to study various aspects of Artin groups: K(π, 1)-conjecture [6,21], acylindrical hyperbolicity [7,18,24], Tits alternative [17], etc. More recently, using the connections between braid groups and mapping class groups, the irreducible parabolic subgroups have been used to define a possible analogue of the complex of curves for Artin groups of spherical type [9,19].…”

Automorphisms of large-type free-of-infinity Artin groups.

“…), proving any property in full generality has remained exceedingly complicated. In general, such conjectures are solved for more specific classes of Artin groups, assuming additional properties about their presentation graphs ( [CD95], [Vas21], [HO19], [Hae19]).…”