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 $I$ be an index set, ${F}_{I}$ the free group generated by $I$
and ${F}_{I}^{+}$ the free semigroup generated by $I$ inside
${F}_{I}\text{.}$ In the following we drop the subscript $I$ when it is clear from the context.

We call the elements of ${F}_{I}$ words and the elements of ${F}_{I}^{+}$ positive words. The empty word is the identity element of ${F}_{I}^{+}\text{.}$ The positive words have unique representations as products of elements of $I$ and the number of factors is the length $L$ of a positive word. The elements of $I$ are called letters. Frequently we shall denote the letters $i\in I$ with the more practical notation ${a}_{i}$ often also with $a,$ $b,$ $c,$ etc. The equivalence relation on positive words $A$ and $B$ is called letterwise agreement and denoted by $A\equiv B\text{.}$

In the following we will very often consider positive words with factors beginning with $a$ and in which only letters $a$ and $b$ occur. Such a word of length $q$ will be denoted ${\u27e8ab\u27e9}^{q}$ so that $${\u27e8ab\u27e9}^{q}\equiv \underset{\underset{q\hspace{0.17em}\text{factors}}{\u23df}}{aba\cdots}$$

**1.2.** Let $M={\left({m}_{ij}\right)}_{i,j\in I}$
be a Coxeter matrix on $I\text{.}$ The *Artin group* ${G}_{M}$ corresponding to
$M$ is the quotient of ${F}_{I}$ by the smallest normal subgroup generated by the relations
${\u27e8ab\u27e9}^{{m}_{ab}}{\left({\u27e8ba\u27e9}^{{m}_{ab}}\right)}^{-1}$
where $a,b\in I$ and ${m}_{ab}\ne \infty \text{.}$
In other words: The Artin group ${G}_{M}$ corresponding to $M$ is the group with generators
${a}_{i},$ $i\in I,$ and the relations
$${\u27e8{a}_{i}{a}_{j}\u27e9}^{{m}_{ij}}={\u27e8{a}_{j}{a}_{i}\u27e9}^{{m}_{ij}}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}i,j\in I,\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{m}_{ij}\ne \infty \text{.}$$
When the Coxeter matrix $M$ is clear from the context we drop the index $M\text{.}$ We denote the
images of the letters and words under the quotient homomorphism
$${F}_{I}\u27f6{G}_{M}$$
by the same symbols and the equivalence relation on elements $A$ and $B$ in ${G}_{M}$
is denoted by $A=B\text{.}$

If $M$ is a Coxeter matrix on $I$ then the *Coxeter group* ${\stackrel{\u203e}{G}}_{M}$
corresponding to $M$ is the group given by generators ${a}_{i},$
$i\in I$ and the relations
$$\begin{array}{ccc}{a}_{i}^{2}& =& 1\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}i\in I,\\ {\u27e8{a}_{i}{a}_{j}\u27e9}^{{m}_{ij}}& =& {\u27e8{a}_{j}{a}_{i}\u27e9}^{{m}_{ij}}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}i,j\in I\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}{m}_{ij}\ne \infty \text{.}\end{array}$$
Obviously this is the same group as the one which is defined by the generators ${a}_{i},$
$i\in I,$ and the usual relations
$${\left({a}_{i}{a}_{j}\right)}^{{m}_{ij}}=1,\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}i,j\in I\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{m}_{ij}\ne \infty \text{.}$$
The images of the elements $A$ of ${G}_{M}$ under the canonical homomorphism
$${G}_{M}\u27f6{\stackrel{\u203e}{G}}_{M}$$
are denoted by $\stackrel{\u203e}{A}$ and the generating system
${\left\{{\stackrel{\u203e}{a}}_{i}\right\}}_{i\in I}$
by $\stackrel{\u203e}{I}\text{.}$ The pair
$({\stackrel{\u203e}{G}}_{M},\stackrel{\u203e}{I})$
is a *Coxeter System* in the sense of Bourbaki [Bou1968] IV 1.3.

In order to describe the Coxeter matrix $M$ we occasionally use the *Coxeter graph* ${\Gamma}_{M}$
in the sense of Bourbaki [Bou1968] IV 1.9.

An Artin group ${G}_{M}$ is of *finite type* resp. *irreducible* resp. *of type*
${A}_{n},$ ${B}_{n},$
${C}_{n},$ ${D}_{n}$ etc, when the Coxeter system
$({\stackrel{\u203e}{G}}_{M},\stackrel{\u203e}{I})$
is finite resp. irreducible resp. of type ${A}_{n},$
${B}_{n},$ ${C}_{n},$
${D}_{n}$ etc.

**1.3.** Let $M$ be a Coxeter matrix. An *elementary transformation* of positive words is a transformation of the form
$$A{\u27e8ab\u27e9}^{{m}_{ab}}B\u27f6A{\u27e8ba\u27e9}^{{m}_{ab}}B$$
where $A,B\in {F}^{+}$ and
$a,b\in I\text{.}$ A *positive transformation of length*
$t$ from a positive word $V$ to a positive word $W$ is a composition of $t$
elementary transformations that begins with $V$ and ends at $W\text{.}$ Two words are
*positive equivalent* if there is a positive transformation that takes one into the other. We indicate positive equivalence of $V$ and
$W$ by $V\u2a66W\text{.}$

The semigroup of positive equivalence classes of positive words relative to $M$ is denoted ${G}_{M}^{+}\text{.}$ The quotient homomorphism ${F}^{+}\u27f6{G}_{M}^{+}$ factors over natural homomorphisms: $${F}^{+}\u27f6{G}_{M}^{+}\u27f6{G}_{M},$$ and for ${G}_{M}$ of finite type we will show that ${G}_{M}^{+}\u27f6{G}_{M}$ is injective. The equivalence relation on elements $V,W\in {G}_{M}^{+}$ is denoted $V\u2a66W\text{.}$

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.