Quasi-projectivity of even Artin groups

Blasco-García, Rubén; Cogolludo-Agustín, José Ignacio

doi:10.2140/gt.2018.22.3979

Cited by 9 publications

(10 citation statements)

References 23 publications

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“…Our first technical lemma is equivalent to Proposition 4.5 (1) in [9]. However, we will include a proof to introduce some of the techniques that we will use in the rest of the paper.…”

Section: Technical Results About Geodesic Words In Large Artin Groupsmentioning

confidence: 99%

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Poly-freeness in large even Artin groups

Blasco-García

2021

Journal of Pure and Applied Algebra

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“…Our first technical lemma is equivalent to Proposition 4.5 (1) in [9]. However, we will include a proof to introduce some of the techniques that we will use in the rest of the paper.…”

Section: Technical Results About Geodesic Words In Large Artin Groupsmentioning

confidence: 99%

“…An Artin group is called even if m e is an even number for all e ∈ E. There are not many results in the literature about even Artin groups (but see [3], [2] and [1]). However, we think that this family deserves more attention since they have remarkable properties some of them shared with the familiy of right-angled Artin groups.…”

Section: Introductionmentioning

confidence: 99%

Poly-freeness in large even Artin groups

Blasco-García

2021

Journal of Pure and Applied Algebra

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“…The fundamental group of the complement of the smooth conic and three tangent lines is the Artin group of the triangle T (4, 4, 2) (i.e. [9]). However, for our purposes, it is more suitable to use a presentation with a more geometrical interpretation.…”

Section: Some Rational Cuspidal Curves On Weighted Projective Planesmentioning

confidence: 99%

Cremona transformations of weighted projective planes, Zariski pairs, and rational cuspidal curves

Bartolo¹,

Cogolludo-Agustín²,

Martín-Morales³

2020

Preprint

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In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to compute the fundamental groups of their complements, using the blow-up-down decompositions of the Cremona transformations, we find examples of Zariski pairs in weighted projective planes (distinguished by the Alexander polynomial). As another application of this machinery we study a family of singularities called weighted Lê-Yomdin, which provide infinite examples of surface singularities with a rational homology sphere link. To end this paper we also study a family of surface singularities generalizing Brieskorn-Pham singularities in a different direction. This family contains infinitely many new examples of integral homology sphere links, answering a question by Némethi.

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“…As before, A Γ is an even Artin group of FC-type, we assume Γ is connected (by Lemma 2.5) but not a clique. We will make use of the following retraction map that works for any even Artin groups, see for example [4,p.311 Remark (2)].…”

Section: 1]mentioning

confidence: 99%

“…Recently, there has been a growing interest in studying even Artin groups, see for example [4,5,6]. In particular, R. Blasco-Garcia, C. Martínez-Pérez and L. Paris proved in [5] that even Artin groups of FC-type are normally poly-free.…”

Section: Introductionmentioning

confidence: 99%

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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Quasi-projectivity of even Artin groups

Cited by 9 publications

References 23 publications

Poly-freeness in large even Artin groups

Poly-freeness in large even Artin groups

Cremona transformations of weighted projective planes, Zariski pairs, and rational cuspidal curves

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

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