Tits Alternative for groups acting properly on $2$-dimensional recurrent complexes (with an appendix written jointly with Jon McCammond)

Osajda, Damian; Przytycki, Piotr

doi:10.48550/arxiv.1904.07796

Cited by 3 publications

(3 citation statements)

References 7 publications

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“…Note that our main Theorem also allows us to recover from a unified perspective several other known results for these groups, such as the above-mentioned Tits alternative (see [44,Corollary B]; this was proved in the extra-large case in [49]), the solubility of the conjugacy problem (also proved in [36,Corollary 1.3] for any two-dimensional Artin group), and control over their quasi-flats (see [35,Theorem 1.1] for the statement about two-dimensional Artin groups, and the closely-related [10, Theorem A] about HHGs). Moreover, we recover the fact that these groups are semi-hyperbolic, as they are coarsely Helly by [31, Thm.…”

Section: Question Given What Is (A Good Bound On) the Asymptotic Dim...mentioning

confidence: 58%

Extra-large type Artin groups are hierarchically hyperbolic

2022

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Section: Question Given What Is (A Good Bound On) the Asymptotic Dim...mentioning

confidence: 58%

Extra-large type Artin groups are hierarchically hyperbolic

2022

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“…Given Γ, what is (a good bound on) the asymptotic dimension of the coned-off Deligne complex? Note that our main Theorem also allows us to recover from a unified perspective several other known results for these groups, such as the above-mentioned Tits alternative (see [MP21,Corollary B]; this was proved in the extra-large case in [OP19]), the solubility of the conjugacy problem (also proved in [HO19, Corollary 1.3] for any two-dimensional Artin group), and control over their quasi-flats (see [HO17,Theorem 1.1] for the statement about two-dimensional Artin groups, and the closely-related [BHS21, Theorem A] about HHGs). Moreover, we recover the fact that these groups are semi-hyperbolic, as they are coarsely Helly by [HHP20, Thm.…”

Section: Introduction Hyperbolic Features Of Artin Groupsmentioning

confidence: 57%

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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“…In particular, it is conjectured that all CAT(0) groups satisfy the Tits alternative. Several classes of CAT(0) groups have been shown to satisfy this alternative, in particular cocompactly cubulated groups [SW05], and groups acting geometrically on two-dimensional systolic complexes or buildings [OP19], but the problem remains open in general.…”

Section: Statement Of Resultsmentioning

confidence: 99%

Acylindrical actions for two-dimensional Artin groups of hyperbolic type

Martin¹,

Przytycki²

2019

Preprint

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For a two-dimensional Artin group A whose associated Coxeter group is hyperbolic, we prove that the action of A on the hyperbolic space obtained by coning off certain subcomplexes of its modified Deligne complex is acylindrical and universal. As a consequence, we obtain the Tits alternative for A, and we classify its subgroups that virtually split as a direct product. We also extend the classification of maximal virtually abelian subgroups to arbitrary two-dimensional Artin groups. A key ingredient in our approach is a simple criterion to show the acylindricity of an action on a two-dimensional CAT(−1) complex.

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Tits Alternative for groups acting properly on $2$-dimensional recurrent complexes (with an appendix written jointly with Jon McCammond)

Cited by 3 publications

References 7 publications

Extra-large type Artin groups are hierarchically hyperbolic

Extra-large type Artin groups are hierarchically hyperbolic

Extra-large type Artin groups are hierarchically hyperbolic

Acylindrical actions for two-dimensional Artin groups of hyperbolic type

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