scholarly journals Braid groups and Artin groups

2009 ◽  
pp. 389-451 ◽  
Author(s):  
Luis Paris
Keyword(s):  
Braid Groups ◽  
Artin Groups

scholarly journals Right-angled Artin groups as normal subgroups of mapping class groups

2021 ◽  
Vol 157 (8) ◽  
pp. 1807-1852
Author(s):  
Matt Clay ◽  
Johanna Mangahas ◽  
Dan Margalit
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Artin Groups ◽  
Normal Subgroups ◽  
Class Groups ◽  
Right Angled Artin Groups ◽  
Proper Normal Subgroup

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

scholarly journals Embedding right-angled Artin groups into graph braid groups

2006 ◽  
Vol 124 (1) ◽  
pp. 191-198 ◽  
Author(s):  
Lucas Sabalka
Keyword(s):  
Braid Groups ◽  
Artin Groups ◽  
Right Angled Artin Groups

scholarly journals Graph braid groups and right-angled Artin groups

2012 ◽  
Vol 364 (1) ◽  
pp. 309-360 ◽  
Author(s):  
Jee Hyoun Kim ◽  
Ki Hyoung Ko ◽  
Hyo Won Park
Keyword(s):  
Braid Groups ◽  
Artin Groups ◽  
Right Angled Artin Groups

scholarly journals On parabolic submonoids of a class of singular Artin monoids

2007 ◽  
Vol 82 (1) ◽  
pp. 29-37
Author(s):  
Noelle Antony
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Knot Theory ◽  
Braid Groups ◽  
Artin Groups ◽  
Parabolic Subgroups ◽  
Vassiliev Invariants ◽  
Sole Purpose

AbstractThis paper concerns parabolic submonoids of a class of monoids known as singular Artin monoids. The latter class includes the singular braid monoid— a geometric extension of the braid group, which was created for the sole purpose of studying Vassiliev invariants in knot theory. However, those monoids may also be construed (and indeed, are defined) as a formal extension of Artin groups which, in turn, naturally generalise braid groups. It is the case, by van der Lek and Paris, that standard parabolic subgroups of Artin groups are canonically isomorphic to Artin groups. This naturally invites us to consider whether the same holds for parabolic submonoids of singular Artin monoids. We show that it is in fact true when the corresponding Coxeter matrix is of ‘type FC’ hence generalising Corran's result in the ‘finite type’ case.

scholarly journals On braid groups and right-angled Artin groups

2014 ◽  
Vol 172 (1) ◽  
pp. 179-190 ◽  
Author(s):  
Francis Connolly ◽  
Margaret Doig
Keyword(s):  
Braid Groups ◽  
Artin Groups ◽  
Right Angled Artin Groups

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

2009 ◽  
Vol 18 (05) ◽  
pp. 651-704 ◽  
Author(s):  
DACIBERG LIMA GONÇALVES ◽  
JOHN GUASCHI
Keyword(s):  
Free Group ◽  
Commutator Subgroup ◽  
Lower Central Series ◽  
Exceptional Case ◽  
Braid Groups ◽  
Central Series ◽  
Artin Groups ◽  
Almost Periodic ◽  
Algebraic Equations ◽  
Derived Series

Motivated in part by the study of Fadell–Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n ≥ 1, the class of m-string braid groups Bm(𝕊2\{x1,…,xn}) of the n-punctured sphere includes the usual Artin braid groups Bm (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type [Formula: see text] (for n = 3). We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of Bm is completely determined for all m ∈ ℕ (respectively, for all m ≠ 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients. When n ≥ 2, we prove that the lower central series (respectively, derived series) of Bm(𝕊2\{x1,…,xn}) is constant from the commutator subgroup onwards for all m ≥ 3 (respectively, m ≥ 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B2(𝕊2\{x1,x2}) admits various interpretations, as the Baumslag–Solitar group BS(2,2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B2(𝕊2\{x1,x2}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product ℤ2 * ℤ. Further, its lower central series quotients Γi/Γi + 1 are direct sums of copies of ℤ2, the number of summands being determined explicitly. In the case m ≥ 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Γi/Γi + 1 are 2-elementary finitely-generated groups.

RANDOMIZED METHODS IN ARTIN GROUPS OF FINITE TYPE

2007 ◽  
Vol 17 (04) ◽  
pp. 859-868
Author(s):  
ROBERT W. BRADSHAW ◽  
STEPHEN P. HUMPHRIES
Keyword(s):  
Finite Type ◽  
Braid Groups ◽  
Artin Groups ◽  
Randomized Methods

We investigate the use of random algorithms for the solution of the conjugacy problems for certain types of Artin groups of finite type including the braid groups.

scholarly journals Helly meets Garside and Artin

2021 ◽  
Author(s):  
Jingyin Huang ◽  
Damian Osajda
Keyword(s):  
Nonpositive Curvature ◽  
Braid Groups ◽  
Fundamental Groups ◽  
Artin Groups ◽  
Dehn Functions ◽  
Complex Reflection Groups ◽  
Helly Property ◽  
Trivial Center ◽  
One Relator Groups ◽  
Arrangements Of Hyperplanes

AbstractA graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a means of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.

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