Braid Pictures for Artin Groups
نویسنده
چکیده
We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams An, Bn = Cn and Dn and the affine diagrams Ãn, B̃n, C̃n and D̃n as subgroups of the braid groups of various simple orbifolds. The cases Dn, B̃n, C̃n and D̃n are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except Ãn the quotient is finite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type Dn.
منابع مشابه
Artin Braid Groups and Homotopy Groups
We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as a summand of the center of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. T...
متن کاملLocal indicability and commutator subgroups of Artin groups
Artin groups (also known as Artin-Tits groups) are generalizations of Artin’s braid groups. This paper concerns Artin groups of spherical type, that is, those whose corresponding Coxeter group is finite, as is the case for the braid groups. We compute presentations for the commutator subgroups of the irreducible spherical-type Artin groups, generalizing the work of Gorin and Lin [GL69] on the b...
متن کاملConjugacy problem for braid groups and Garside groups
We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee [3]. This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among others).
متن کاملEmbeddings of graph braid and surface groups in right-angled Artin groups and braid groups
We prove by explicit construction that graph braid groups and most surface groups can be embedded in a natural way in right-angled Artin groups, and we point out some consequences of these embedding results. We also show that every right-angled Artin group can be embedded in a pure surface braid group. On the other hand, by generalising to rightangled Artin groups a result of Lyndon for free gr...
متن کاملLecture Notes on Artin–tits Groups
1 0.1. Braid groups 1 0.2. Artin–Tits groups 2 1. The general case 2 1.1. The word reversing technique 3 1.2. Artin–Tits monoids 4 1.3. Artin–Tits groups 5 1.4. Exercises 5 2. The spherical case 6 2.1. Background about Coxeter groups 6 2.2. Garside structure 7 2.3. Normal form 7 2.4. Exercises 8 3. The braid case 9 3.1. The Artin representation 10 3.2. Handle reduction 11 3.3. The braid orderin...
متن کامل