Last update: 26 January 2014
In these paragraphs we shall define the Artin groups, and fix some of the notations and ideas which will follow.
1.1. Let be an index set, the free group generated by and the free semigroup generated by inside In the following we drop the subscript when it is clear from the context.
We call the elements of words and the elements of positive words. The empty word is the identity element of The positive words have unique representations as products of elements of and the number of factors is the length of a positive word. The elements of are called letters. Frequently we shall denote the letters with the more practical notation often also with etc. The equivalence relation on positive words and is called letterwise agreement and denoted by
In the following we will very often consider positive words with factors beginning with and in which only letters and occur. Such a word of length will be denoted so that
1.2. Let be a Coxeter matrix on The Artin group corresponding to is the quotient of by the smallest normal subgroup generated by the relations where and In other words: The Artin group corresponding to is the group with generators and the relations When the Coxeter matrix is clear from the context we drop the index We denote the images of the letters and words under the quotient homomorphism by the same symbols and the equivalence relation on elements and in is denoted by
If is a Coxeter matrix on then the Coxeter group corresponding to is the group given by generators and the relations Obviously this is the same group as the one which is defined by the generators and the usual relations The images of the elements of under the canonical homomorphism are denoted by and the generating system by The pair is a Coxeter System in the sense of Bourbaki [Bou1968] IV 1.3.
In order to describe the Coxeter matrix we occasionally use the Coxeter graph in the sense of Bourbaki [Bou1968] IV 1.9.
An Artin group is of finite type resp. irreducible resp. of type etc, when the Coxeter system is finite resp. irreducible resp. of type etc.
1.3. Let be a Coxeter matrix. An elementary transformation of positive words is a transformation of the form where and A positive transformation of length from a positive word to a positive word is a composition of elementary transformations that begins with and ends at Two words are positive equivalent if there is a positive transformation that takes one into the other. We indicate positive equivalence of and by
The semigroup of positive equivalence classes of positive words relative to is denoted The quotient homomorphism factors over natural homomorphisms: and for of finite type we will show that is injective. The equivalence relation on elements is denoted
This is a translation, with notes, of the paper, Artin-Gruppen und Coxeter-Gruppen, Inventiones math. 17, 245-271, (1972).
Translated by: C. Coleman, R. Corran, J. Crisp, D. Easdown, R. Howlett, D. Jackson and A. Ram at the University of Sydney, 1996.