XXL type Artin groups are CAT(0) and acylindrically hyperbolic

Haettel, Thomas

doi:10.48550/arxiv.1905.11032

Cited by 10 publications

(13 citation statements)

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“…The only exception we have now is (2,4,4) If A Γ is a clique with all edges labels at least 5, it is CAT(0) by [18]. The fact that CAT(0) groups satisfy FJCw is proved in [1,26].…”

Section: 1]mentioning

confidence: 96%

Poly-freeness of Artin groups and the Farrell-Jones Conjecture

Wu¹

2019

Preprint

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We give two simple proofs of the fact that any even Artin groups of FC-type are normally poly-free which was recently established by R. Blasco-Garcia, C. Martínez-Pérez and L. Paris. More generally, let Γ be a finite simplicial graph with all edges labelled by positive even integers and A Γ be its associated Artin group, we show that if A T is poly-free (resp. normally poly-free) for every clique T in Γ, then A Γ is poly-free (resp. normally poly-free). We also prove similar results regarding the Farrell-Jones Conjecture for Artin groups. In particular, we show that if A Γ is an even Artin group such that each clique in Γ either has at most 3 vertices, has all of its labels at least 6, or is the join of these two types of cliques (the edges connecting the cliques are all labelled by 2), then A Γ satisfies the Farrell-Jones Conjecture. We also have some results for general Artin groups. of G. Our first theorem can be stated as follows.Theorem A. Let Γ be a finite simplicial graph with all edges labelled by positive even integers and A Γ be its associated Artin group. If A Γ is of FC-type, then it is normally polyfree. In general if A T is poly-free (resp. normally poly-free) for every clique T in Γ, then A Γ is poly-free (resp. normally poly-free).In [3, Question 2] and the discussions below it, Bestvina asks whether all Artin groups are virtually poly-free. In light of our main theorem, one might not need to pass to a finite index subgroup in the case of even Artin groups although our reduction does also work for

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“…The only exception we have now is (2,4,4) If A Γ is a clique with all edges labels at least 5, it is CAT(0) by [18]. The fact that CAT(0) groups satisfy FJCw is proved in [1,26].…”

Section: 1]mentioning

confidence: 96%

Poly-freeness of Artin groups and the Farrell-Jones Conjecture

Wu¹

2019

Preprint

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“…Concerning the Baum-Connes conjecture, the only examples were essentially braid groups (see [OO01] and [Sch07]), some large type Artin groups (see [CHR16]) and XXL type Artin groups (see [Hae19]).…”

Section: Corollary Bmentioning

confidence: 99%

“…Concerning CAT(0) spaces, R. Charney asks whether every Artin-Tits group acts properly and cocompactly on a CAT(0) space (see [Cha]). Very few cases are known, essentially right-angled Artin groups (see [CD95]), groups with few generators (see [Bra00], [BM10], [HKS16]) and groups with sufficiently large labels (see [BC02], [BM00], [Bel05], [Hae19]).…”

Section: Introductionmentioning

confidence: 99%

Cubulation of some triangle-free Artin groups

Haettel¹

2020

Preprint

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We prove that some classes of triangle-free Artin-Tits groups act properly on locally finite, finite-dimensional CAT(0) cube complexes. In particular, this provides the first examples of Artin-Tits groups that are properly cubulated but cannot be cocompactly cubulated, even virtually. The existence of such a proper action has many interesting consequences for the group, notably the Haagerup property, and the Baum-Connes conjecture with coefficients.

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“…For instance, Conjecture 1.2. (1) is known to hold for right angled Artin groups ( [CD95]), some classes of 2-dimensional Artin groups ( [BC02], [BM00], [Hae19]) or spherical Artin groups of rank 3 ([B + 00]). As regards to Conjecture 1.2.…”

Section: Introductionmentioning

confidence: 99%

“…In [CMW19], Charney and Morris-Wright also showed that Artin groups A Γ whose defining graph Γ is not a join are acylindrically hyperbolic, extending the result of Chatterji and Martin for Artin groups of type FC whose defining graph has diameter at most 3 ([CM19]). It is also known from [Hae19] that XXL Artin groups (those with coefficient at least 5) of rank at least 3 are acylindrically hyperbolic. More recently, Kato and Oguni showed that triangle-free Artin groups and Artin groups of large type associated to cones over square-free bipartite graphs are also acylindrically hyperbolic ([KO20]).…”

Section: Introductionmentioning

confidence: 99%