Acylindrical hyperbolicity for Artin groups of dimension 2

Vaskou, Nicolas

doi:10.48550/arxiv.2007.16169

Cited by 2 publications

(3 citation statements)

References 16 publications

(24 reference statements)

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“…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 (Charney & Davis, 1995), and have been used to study various aspects of Artin groups: K(π, 1)-conjecture (Charney & Davis, 1995, Paris, 2014, acylindrical hyperbolicity (Charney & Morris-Wright, 2019, Martin & Przytycki, 2019, Vaskou, 2020, Tits alternative (Martin & Przytycki, 2020), 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 (Cumplido et al, 2019, Morris-Wright, 2021.…”

Section: Introductionmentioning

confidence: 99%

Parabolic subgroups of large-type Artin groups

Cumplido¹,

Martin²,

Vaskou³

2020

Preprint

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We show that the geometric realisation of the poset of proper parabolic subgroups of a large-type Artin group has a systolic geometry. We use this geometry to show that the set of parabolic subgroups of a large-type Artin group is stable under arbitrary intersections and forms a lattice for the inclusion. As an application, we show that parabolic subgroups of large-type Artin groups are stable under taking roots and we completely characterise the parabolic subgroups that are conjugacy stable.We also use this geometric perspective to recover and unify results describing the normalizers of parabolic subgroups of large-type Artin groups.

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Section: Introductionmentioning

confidence: 99%

Parabolic subgroups of large-type Artin groups

Cumplido¹,

Martin²,

Vaskou³

2020

Preprint

Self Cite

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“…The action of A ab on Lk D pv ab q has been studied by Vaskou in [Vas20]. In particular, the following result, which is a geometric counterpart of Lemma 2.6 will be useful in Section 5: Lemma 2.34 ([Vas20, Proposition 4.7]).…”

Section: 4mentioning

confidence: 99%

“…This question has been answered positively for most standard classes of Artin groups, such as right-angled Artin groups [CS11], Artin groups of finite type [CW:], Artin groups of Euclidean type [Cal20], Artin groups whose underlying presentation graph is not a join [RMW19], and two-dimensional Artin groups [Vas20].…”

Section: Introduction Hyperbolic Features Of Artin Groupsmentioning

confidence: 99%

Extra-large type Artin groups are hierarchically hyperbolic

Hagen¹,

Martin²,

Sisto³

2021

Preprint

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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

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Acylindrical hyperbolicity for Artin groups of dimension 2

Cited by 2 publications

References 16 publications

Parabolic subgroups of large-type Artin groups

Parabolic subgroups of large-type Artin groups

Extra-large type Artin groups are hierarchically hyperbolic

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