Artin groups and Coxeter groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 26 January 2014

Introduction

An Artin group is a group $G$ with a presentation by a system of generators $a_i,$ $i\in I,$ and relations $$a_i{a}_j{a}_i\cdots =a_j{a}_i{a}_j,i,j\in I$$ where the words on each side of these relations are sequences of $m_{ij}$ letters where $a_i$ and $a_j$ alternate in the sequence. The matrix of values $m_{ij}$ is a Coxeter matrix $M={\left(m_{ij}\right)}_{i,j\in I}$ on $I\text{.}$ These groups generalize the braid groups established in 1925 by E. Artin in a natural way and therefore we suggest naming them Artin groups.

If one adds the relations $a_i^2=1$ to the relations in the presentation of an Artin group then one gets a presentation of a Coxeter group $\bar{G}\text{.}$ Thus the Coxeter groups are quotient groups of the Artin groups. It is well known that in the case of the braid group one gets the symmetric group in this way.

Since their introduction by Coxeter in 1935 the Coxeter groups have been well studied and a nice presentation of the results can be found in Bourbaki [Bou1968]. Other than the free groups, the braid group is the only class of Artin groups that has had a serious line of investigation, in particular, recently the solution of the conjugation problem was given by Garside. For the other Artin groups, a few isolated results appear in [Bri1971], [Bri1972] and [Gar1969]. These references, as well as our own work here, concentrate, for the most part, on the case that the Artin group $G$ corresponds to a finite Coxeter group. The Coxeter groups were already classified by Coxeter himself: these are the finite reflection groups - the irreducible cases being, the groups of Types $A_n,$ $B_n,$ $C_n,$ $D_n,$ $E_6,$ $E_7,$ $E_8,$ $F_4,$ $G_2,$ $H_3,$ $H_4$ and $I_2\left(p\right)$ with $p=5$ or $p\ge 7$ (see [Bou1968] VI §4.1). It was proved in [Bri1971] that for these finite reflection groups the Artin group $G$ is the fundamental group of the spaces $X_G$ of regular orbits for which $G$ is the corresponding complex reflection group. In [Bri1972] we conjectured, and for a few cases proved, that $X_G$ is an Eilenberg-McLane space so that the cohomology of $X_G$ is isomorphic to the cohomology of $G,$ and thus a few statements were proved about the cohomology of $G\text{.}$

In the following work we study the Artin groups by combinatorial methods, which are very similar to those of Garside. For $G$ with finite $\bar{G}$ we solve the word problem and the conjugation problem and we determine the centre of $G\text{.}$ For irreducible $\bar{G}$ the centre of $G$ is infinite cyclic and generated by an appropriate power of the product $a_i\cdots a_n$ of the generators of $G\text{.}$ For some cases these results were already known, and J.P. Serre asked us whether this was always the case. This question was the starting point of our work and we would like to thank J.P. Serre for his direction.

Deligne told us that he had constructed in the manner of Tits simplicial complexes on which $G$ operates, and has proved that $X_G$ is an Eilenberg-McLane space for all $G$ with finite $\bar{G}\text{.}$ We hope that our own work is not made superfluous by the very interesting work of Deligne, which we have not yet seen.

Notes and references

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.

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