Hierarchically hyperbolic groups and uniform exponential growth

Abbott, Carolyn R.; Ng, Tony; Davide, Spriano,; Gupta, Radhika; Petyt, Harry

doi:10.48550/arxiv.1909.00439

Cited by 10 publications

(14 citation statements)

References 29 publications

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“…We now handle assertion (2) (uniform exponential growth). In the case where A Γ is not dihedral, we have that A Γ is torsion-free by [CD95] and acylindrically hyperbolic by [MP21], so by Corollary 1.3 of [ANS19], A Γ has uniform exponential growth. The remaining case is where A Γ is dihedral, i.e.…”

Section: Proof Ofmentioning

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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Section: Proof Ofmentioning

confidence: 99%

Extra-large type Artin groups are hierarchically hyperbolic

Hagen¹,

Martin²,

Sisto³

2021

Preprint

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“…Proposition 3.2 establishes a sufficient condition in terms of separability of stabilisers of product regions for an HHG to virtually admit a BBF colouring. For the purposes of Theorem 3.15, a virtual BBF colouring is sufficient, since finite-index subgroups of HHGs are HHGs with the same hierarchically hyperbolic structure [ANS19]. Examples of HHGs with virtual BBF colourings (also known as colourable HHGs [DMS20]) include: ‚ hyperbolic groups; ‚ mapping class groups of finite-type oriented hyperbolic surfaces [BHS19,BBF15]; ‚ virtually compact special groups (e.g.…”

Section: Introductionmentioning

confidence: 99%

Projection complexes and quasimedian maps

Hagen¹,

Petyt²

2021

Preprint

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We use the projection complex machinery of Bestvina-Bromberg-Fujiwara to study hierarchically hyperbolic groups. In particular, we show that if the group has a BBF colouring and its associated hyperbolic spaces are quasiisometric to trees, then the group is quasiisometric to a finite-dimensional CAT(0) cube complex. We deduce various properties, including the Helly property for hierarchically quasiconvex subsets.

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“…Hierarchically hyperbolic groups are known to satisfy a number of other properties such as having finite asymptotic dimension [BHS17b, Theorem A], having a uniform bound on the conjugator length of Morse elements [AB19], and for virtually torsion-free HHGs, their uniform exponential growth is well understood [ANS19].…”

Section: Introductionmentioning

confidence: 99%