Introduction to Teichmüller theory, old and new, II

24 Athanase Papadopoulos
In Chapter 11 of this volume, the theory of braids is included in a very wide setting
that encompasses mapping class groups, but also other combinatorially defined finitely
presented groups, namely Garside groups, Artin groups and Coxeter groups. To make
things more precise, we take a finite set S of cardinality n and we recall that a Coxeter
matrix over S is an n × n matrix whose coefficients mst (s, t ∈ S) belong to the set
{1, 2,...,∞}, with mst = 1 if and only if s = t. The Coxeter graph Ɣ associated
to a Coxeter matrix M = ms,t is a labeled graph whose vertex set is S and where
two distinct vertices s and t are joined by an edge whenever ms,t ≥ 3. If mst ≥ 4,
then the edge is labeled by ms,t. Coxeter graphs are also called Dynkin diagrams.
The Coxeter group of type Ɣ is the finitely presented group with generating set S and
relations s 2 = 1 for s in S, and (st) mst = 1 for s = t in S. Here, a relation with
mst =∞means that the relation does not exist.
The Artin group associated to a Coxeter matrix M = ms,t is a group defined by
generators and relations, where the generators are the elements of S, ordered as a sequence
{a1,...,an} and where the relations are defined by the equalities 〈a1,a2〉 m1,2 =
〈a2,a1〉 m2,1,...,〈an−1,an〉 mn−1,n =〈an,an−1〉 mn,n−1 for all mi,j ∈{2, 3,...,∞},
where 〈ai,aj 〉 denotes the alternating product of ai and aj taken mi,j times, starting
with ai. (For example, 〈a1,a2〉 5 = a1a2a1a2a1.) Artin groups are also used in other
domains of mathematics, for instance in the theory of random walks.
Coxeter groups were introduced by J. Tits in relation with his study ofArtin groups.
Garside groups were introduced by P. Dehornoy and L. Paris, as a generalization of
Artin groups. There are several relations between Artin groups, Coxeter groups and
Garside groups. One important aspect of Garside groups is that these groups are wellsuited
to the study of algorithmic problems for braid groups. An Artin group has a
quotient Coxeter group.
There is a geometric interpretation of Artin groups which extends the interpretation
of braid groups in terms of fundamental groups of hyperplane arrangements in C n .Itis
unknown whether mapping class groups areArtin groups and whether they are Garside
groups. Some Artin groups, called Artin groups of spherical type, are Garside groups,
and it is known that Artin groups of spherical type are generalizations of braid groups.
Chapter 11 contains algebraic results, algorithmic results, and results on the representation
theory of these classes of groups.
From an algebraic point of view, Paris gives an account of known results on the cohomology
of braid groups and ofArtin groups of spherical type. He introduces Salvetti
complexes of hyperplane arrangements. These complexes are simplicial complexes
that arise naturally in the study of hyperplane arrangements; they have natural geometric
realizations, and they have been successfully used as a tool in computing the
cohomology of Artin groups.
From the algorithmic point of view, the author reports on Tits’ solution of the word
problem for Coxeter groups, on Garside’s solution of the conjugacy problem for braid
groups, and on recent progress made by Dehornoy and Paris on the extension of this
result to Garside groups.