Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity

Haettel, Thomas; Hoda, Nima; Petyt, Harry

doi:10.48550/arxiv.2009.14053

Cited by 15 publications

(19 citation statements)

References 46 publications

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“…Birman-Chillingworth [2, Theorem 1] (for closed surfaces), Szepietowski [22, Lemma 3] (for surfaces with boundaries) and Gonçalves-Guaschi-Maldonado [8, Theorem 1.1] (for surfaces with punctures) proved that the mapping class group of a nonorientable surface N b g,p is a subgroup of the mapping class group of the orientation double cover S 2b g−1,2p (we will describe the induced injective homomorphism in Section 3). In this paper we show the following as an application of the semihyperbolicity of the mapping class group of orientable surfaces, independently established by Durham-Minsky-Sisto [6, Corollary D] and Haettel-Hoda-Petyt [9,Corollary 3.11] Theorem 1.1. For all but (g, p, b) = (2, 0, 0), the mapping class group Mod(N b g,p ) is undistorted in the mapping class group Mod(S 2b g−1,2p ).…”

Section: Introductionmentioning

confidence: 75%

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

Katayama¹,

Kuno²

2021

Preprint

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Let N be a connected nonorientable surface with or without boundary and punctures, and j : S → N the orientation double covering. Birman-Chillingworth, Szepietowski, and Gonçalves-Guaschi-Maldonado proved that the orientation double covering j induces an injective homomorphism ι : Mod(N ) ֒→ Mod(S) with one exception. In this paper we prove that this injective homomorphism ι is a quasi-isometric embedding as an application of the semihyperbolicity of Mod(S), which is established by Durham-Minsky-Sisto and Haettel-Hoda-Petyt. We also prove that the embedding Mod(F ′ ) ֒→ Mod(F ) induced by an inclusion of a pair of possibly nonorientable surfaces F ′ ⊂ F , well-studied by Paris-Rolfsen and Stukow, is a quasiisometric embedding.

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Section: Introductionmentioning

confidence: 75%

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

Katayama¹,

Kuno²

2021

Preprint

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“…Their discrete counterpart are called Helly graphs. Their use in geometric group theory is recent and growing, see notably [Lan13], [HO19], [CCG + 20], [HHP20], [OV20], [Hae21]. Roughly speaking, CAT(0) spaces are typically locally Euclidean spaces, whereas injective metric spaces are typically locally ℓ ∞ metric spaces.…”

Section: Introductionmentioning

confidence: 99%

Lattices, injective metrics and the $K(π,1)$ conjecture

Haettel¹

2021

Preprint

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Starting with a lattice with an action of Z or R, we build a Helly graph or an injective metric space. We deduce that the ℓ ∞ orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces. We apply this to show that any Garside group acts on an injective metric space and on a Helly graph. We also deduce that the natural piecewise ℓ ∞ metric on any Euclidean building of type A n extended, B n , C n or D n is injective, and its thickening is a Helly graph. Concerning Artin groups of Euclidean types A n and C n , we show that the natural piecewise ℓ ∞ metric on the Deligne complex is injective, the thickening is a Helly graph, and it admits a convex bicombing. This gives a metric proof of the K(π, 1) conjecture, as well as several other consequences usually known when the Deligne complex has a CAT(0) metric.

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“…Since this notion gives much better control than acylindrical hyperbolicity, it implies many results expected of non-positively curved groups that are not true for more general notions of non-positive curvature; for example, it gives quadratic isoperimetric inequality [Bow13,BHS19], solubility of the word and conjugacy problem [BHS19,HHP20], the Tits alternative [DHS17,DHS20], finite asymptotic dimension [BHS17a], semi-hyperbolicity [HHP20,DMS20], etc. In particular, since both braid groups and right-angled Artin groups belong to this family [BHS17b,BHS19], the following is a natural question that was for instance raised by Calvez-Wiest [CW19]:…”

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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Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity

Cited by 15 publications

References 46 publications

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

Lattices, injective metrics and the $K(π,1)$ conjecture

Extra-large type Artin groups are hierarchically hyperbolic

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