### Citations

670 |
Reflection groups and Coxeter groups. Cambridge
- Humphreys
- 1990
(Show Context)
Citation Context ...bras, Chapters 4, 5, and 6” [5]. They are involved in several areas of mathematics such as group theory, Lie theory, or hyperbolic geometry, and are in some sense fairly well understood. We recommend =-=[22]-=-, [13] and [5] for a detailed study on these groups. Artin-Tits groups were also introduced by Tits [29], as extensions of Coxeter groups. There is no general result on these groups, and the theory co... |

208 |
Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles
- Bourbaki
- 1968
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Citation Context ... group of type Γ. Coxeter groups were introduced by Tits in his manuscript [28], which was used by Bourbaky as a basis for writing his seminal book “Lie groups and Lie algebras, Chapters 4, 5, and 6” =-=[5]-=-. They are involved in several areas of mathematics such as group theory, Lie theory, or hyperbolic geometry, and are in some sense fairly well understood. We recommend [22], [13] and [5] for a detail... |

191 |
immeubles des groupes de tresses généralisés
- Deligne, Les
- 1972
(Show Context)
Citation Context ...s groups, it suffices to consider connected Coxeter graphs. We say that a Coxeter graph Γ is of spherical type if WΓ is finite. It is known that Z(AΓ) ≃ Z if Γ is connected and of spherical type (see =-=[14, 6]-=-), and we believe that the center of any other Artin-Tits group associated to a connected Coxeter graph is trivial. In other words: Conjecture B. Let Γ be a non-spherical connected Coxeter graph. Then... |

157 | The geometry and topology of Coxeter groups
- Davis
- 2008
(Show Context)
Citation Context ...Chapters 4, 5, and 6” [5]. They are involved in several areas of mathematics such as group theory, Lie theory, or hyperbolic geometry, and are in some sense fairly well understood. We recommend [22], =-=[13]-=- and [5] for a detailed study on these groups. Artin-Tits groups were also introduced by Tits [29], as extensions of Coxeter groups. There is no general result on these groups, and the theory consists... |

73 |
Normalisateurs de tores. I. Groupes de Coxeter étendus
- Tits
- 1966
(Show Context)
Citation Context ... Lie theory, or hyperbolic geometry, and are in some sense fairly well understood. We recommend [22], [13] and [5] for a detailed study on these groups. Artin-Tits groups were also introduced by Tits =-=[29]-=-, as extensions of Coxeter groups. There is no general result on these groups, and the theory consists on the study of more or less extended families. In particular, the following basic questions are ... |

64 |
Geodesic automation and growth functions of Artin groups
- Charney
- 1995
(Show Context)
Citation Context ...t 6=∞〉 + . By [6] the natural homomorphism A+ → A is injective (see also [26]). Moreover, by [23], we have A+ ∩ AX = A + X (this equality is also a direct consequence of the normal forms defined in 5 =-=[9]-=-). On the other hand, it is easily deduced from the presentation of A+ that, for a ∈ A+, we have a ∈ A+X if and only if any expression of a in the elements of Σ is actually a word in the elements of Σ... |

56 |
The homotopy type of complex hyperplane complements
- Lek
(Show Context)
Citation Context ...(⋃ r∈R (Hr ×Hr) ) . This is a connected manifold of dimension 2|S| on which the group W acts freely and properly discontinuously. A key result in the domain is the following. Theorem 2.1 (Van der Lek =-=[23]-=-). The fundamental group of MΓ/W is isomorphic to the Artin-Tits group AΓ. Recall that a CW-complex X is a a classifying space for a (discrete) group G if π1(X) = G and the universal cover of X is con... |

52 |
The K(π, 1)-problem for hyperplane complements associated to infinite reflection groups
- Charney, Davis
- 1995
(Show Context)
Citation Context ...of Γ with three or more vertices is of spherical type. After [4] and [21] the next significant step in the study of Artin-Tits groups and, more specifically, on the four conjectures stated above, was =-=[10]-=-. In this paper Conjecture D (and hence Conjecture A) was proved for FC type Artin-Tits groups and 2-dimensional Artin-Tits groups (that include the Artin-Tits groups of large type). Later, Conjecture... |

33 | Braid pictures for Artin groups
- Allcock
(Show Context)
Citation Context ...is affine. As far as we know there is no known explicit algorithm for solving the word problem for these groups, except for the groups of type Ãn and C̃n (see [15, 16]). The techniques introduced in =-=[1]-=- may be used to solve the question for the groups of type B̃n and D̃n, but we have no idea on how to treat the groups of type Ẽk, k = 6, 7, 8. The K(π, 1) conjecture was proved for the groups of type... |

33 | Artin monoids inject in their group
- Paris
(Show Context)
Citation Context ...Tits monoid of Γ is defined by the monoid presentation A+ = 〈Σ | Π(σs, σt : ms,t) = Π(σt, σs : ms,t) , s, t ∈ S, s 6= t and ms,t 6=∞〉 + . By [6] the natural homomorphism A+ → A is injective (see also =-=[26]-=-). Moreover, by [23], we have A+ ∩ AX = A + X (this equality is also a direct consequence of the normal forms defined in 5 [9]). On the other hand, it is easily deduced from the presentation of A+ tha... |

28 |
Artin-Gruppen und Coxeter-Gruppen
- Brieskorn, Saito
- 1972
(Show Context)
Citation Context ...s groups, it suffices to consider connected Coxeter graphs. We say that a Coxeter graph Γ is of spherical type if WΓ is finite. It is known that Z(AΓ) ≃ Z if Γ is connected and of spherical type (see =-=[14, 6]-=-), and we believe that the center of any other Artin-Tits group associated to a connected Coxeter graph is trivial. In other words: Conjecture B. Let Γ be a non-spherical connected Coxeter graph. Then... |

26 |
Artin groups and infinite Coxeter groups
- Appel, Schupp
- 1983
(Show Context)
Citation Context ...e D was proved in [14]. We say that Γ is of large type if ms,t ≥ 3 for all s, t ∈ S, s 6= t, and that Γ is of extra-large type if ms,t ≥ 4 for all s, t ∈ S, s 6= t. Conjectures A and C were proved in =-=[4]-=- for extra-large type ArtinTits groups, and Conjecture B can be easily proved with the same techniques. Conjecture D for large type Artin-Tits groups is a straightforward consequence of [21]. Recall t... |

18 | TheK(π, 1) conjecture for the affine braid groups
- Charney, Peifer
- 2002
(Show Context)
Citation Context ...ion for the groups of type B̃n and D̃n, but we have no idea on how to treat the groups of type Ẽk, k = 6, 7, 8. The K(π, 1) conjecture was proved for the groups of type Ãn and C̃n in [24] (see also =-=[11]-=-) and for the groups of type B̃n in [8]. The other cases are open. On the other hand, none of these papers addresses Conjecture B. 3 From free of infinity Artin-Tits groups to Artin-Tits groups The ai... |

17 |
Présentations duales des groupes de tresses de type affine A
- Digne
(Show Context)
Citation Context ... for which the associated Coxeter group is affine. As far as we know there is no known explicit algorithm for solving the word problem for these groups, except for the groups of type Ãn and C̃n (see =-=[15, 16]-=-). The techniques introduced in [1] may be used to solve the question for the groups of type B̃n and D̃n, but we have no idea on how to treat the groups of type Ẽk, k = 6, 7, 8. The K(π, 1) conjectur... |

12 |
The word problem for Artin groups of FC type
- Altobelli
- 1998
(Show Context)
Citation Context ...ater, Conjecture B was proved for Artin-Tits groups of FC type in [18], and for 2-dimensional Artin-Tits groups in [19]. On the other hand, Conjecture C was proved for Artin-Tits groups of FC type in =-=[2, 3]-=-, and for 2-dimensional ArtinTits groups in [12]. Note also that a new solution to the word problem for FC type Artin-Tits groups will follow from Theorem C. A challenging question in the domain is to... |

11 | Universal cover of Salvetti’s complex and topology of simplicial arrangements of hyperplanes
- Paris
- 1993
(Show Context)
Citation Context ...et ΣX = {σs; s ∈ X}, and we denote by AX the subgroup of A = AΓ generated by ΣX . It is known that (WX ,X) is the Coxeter system of ΓX (see [5]), and that (AX ,ΣX) is the Artin-Tits system of ΓX (see =-=[23, 25, 20]-=-). The subgroup WX is called standard parabolic subgroup of W , and AX is called standard parabolic subgroup of A. The proof of Theorem A is elementary. That of Theorem B is more complicated and requi... |

9 | Parabolic subgroups of Artin groups of type FC
- Godelle
(Show Context)
Citation Context ... A) was proved for FC type Artin-Tits groups and 2-dimensional Artin-Tits groups (that include the Artin-Tits groups of large type). Later, Conjecture B was proved for Artin-Tits groups of FC type in =-=[18]-=-, and for 2-dimensional Artin-Tits groups in [19]. On the other hand, Conjecture C was proved for Artin-Tits groups of FC type in [2, 3], and for 2-dimensional ArtinTits groups in [12]. Note also that... |

8 |
A geometric rational form for Artin groups of FC type
- Altobelli, Charney
(Show Context)
Citation Context ...ater, Conjecture B was proved for Artin-Tits groups of FC type in [18], and for 2-dimensional Artin-Tits groups in [19]. On the other hand, Conjecture C was proved for Artin-Tits groups of FC type in =-=[2, 3]-=-, and for 2-dimensional ArtinTits groups in [12]. Note also that a new solution to the word problem for FC type Artin-Tits groups will follow from Theorem C. A challenging question in the domain is to... |

8 | The k(π, 1)- problem for the affine Artin group of type B̃n and its cohomology
- CALLEGARO, MORONI, et al.
(Show Context)
Citation Context ...but we have no idea on how to treat the groups of type Ẽk, k = 6, 7, 8. The K(π, 1) conjecture was proved for the groups of type Ãn and C̃n in [24] (see also [11]) and for the groups of type B̃n in =-=[8]-=-. The other cases are open. On the other hand, none of these papers addresses Conjecture B. 3 From free of infinity Artin-Tits groups to Artin-Tits groups The aim of this section is to prove the follo... |

8 |
Das K(π, 1)-Problem für die affinen Wurzelsysteme vom Typ
- Okonek
- 1979
(Show Context)
Citation Context ...solve the question for the groups of type B̃n and D̃n, but we have no idea on how to treat the groups of type Ẽk, k = 6, 7, 8. The K(π, 1) conjecture was proved for the groups of type Ãn and C̃n in =-=[24]-=- (see also [11]) and for the groups of type B̃n in [8]. The other cases are open. On the other hand, none of these papers addresses Conjecture B. 3 From free of infinity Artin-Tits groups to Artin-Tit... |

6 |
A Garside presentation for Artin-Tits groups of type C̃n
- Digne
- 2010
(Show Context)
Citation Context ... for which the associated Coxeter group is affine. As far as we know there is no known explicit algorithm for solving the word problem for these groups, except for the groups of type Ãn and C̃n (see =-=[15, 16]-=-). The techniques introduced in [1] may be used to solve the question for the groups of type B̃n and D̃n, but we have no idea on how to treat the groups of type Ẽk, k = 6, 7, 8. The K(π, 1) conjectur... |

6 |
The K(pi, 1) conjecture for a class of Artin groups
- Ellis, Sköldberg
(Show Context)
Citation Context ... if ms,t 6= ∞ for all s, t ∈ S, s 6= t. In Section 3 we prove the following principle on the conjectures presented in Section 2 (except for the one on the cohomology, which has been already proved in =-=[17]-=- and [20]). Principle. If a property is true for all free of infinity Artin-Tits groups, then it will be true for all Artin-Tits groups. A careful reader may point out to the authors that this princip... |

6 |
Hyperplane complements of large type
- HENDRIKS
- 1985
(Show Context)
Citation Context ... proved in [4] for extra-large type ArtinTits groups, and Conjecture B can be easily proved with the same techniques. Conjecture D for large type Artin-Tits groups is a straightforward consequence of =-=[21]-=-. Recall that a Coxeter graph Γ is free of infinity if ms,t 6=∞ for all s, t ∈ S. We say that Γ is of FC type if every free of infinity full subgraph of Γ is of spherical type. On the other hand, we s... |

5 |
Locally non-spherical Artin groups
- Chermak
- 1998
(Show Context)
Citation Context ...s of FC type in [18], and for 2-dimensional Artin-Tits groups in [19]. On the other hand, Conjecture C was proved for Artin-Tits groups of FC type in [2, 3], and for 2-dimensional ArtinTits groups in =-=[12]-=-. Note also that a new solution to the word problem for FC type Artin-Tits groups will follow from Theorem C. A challenging question in the domain is to prove Conjectures A, B, C, and D for the so-cal... |

5 |
Artin-Tits groups with CAT(0) Deligne complex
- Godelle
(Show Context)
Citation Context ...2-dimensional Artin-Tits groups (that include the Artin-Tits groups of large type). Later, Conjecture B was proved for Artin-Tits groups of FC type in [18], and for 2-dimensional Artin-Tits groups in =-=[19]-=-. On the other hand, Conjecture C was proved for Artin-Tits groups of FC type in [2, 3], and for 2-dimensional ArtinTits groups in [12]. Note also that a new solution to the word problem for FC type A... |

5 |
1) and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups
- unknown authors
(Show Context)
Citation Context ...6= ∞ for all s, t ∈ S, s 6= t. In Section 3 we prove the following principle on the conjectures presented in Section 2 (except for the one on the cohomology, which has been already proved in [17] and =-=[20]-=-). Principle. If a property is true for all free of infinity Artin-Tits groups, then it will be true for all Artin-Tits groups. A careful reader may point out to the authors that this principle is fal... |

4 |
Groupes et géométries de Coxeter (Institut des Hautes Études Scientifiques
- Tits
- 1961
(Show Context)
Citation Context ...by the Agence Nationale de la Recherche (projet Théorie de Garside, ANR-08-BLAN-0269-03). 1 The group A is called Artin-Tits group of type Γ. Coxeter groups were introduced by Tits in his manuscript =-=[28]-=-, which was used by Bourbaky as a basis for writing his seminal book “Lie groups and Lie algebras, Chapters 4, 5, and 6” [5]. They are involved in several areas of mathematics such as group theory, Li... |