Last update: 7 May 2013
First we recall some definitions and facts from [Bou1968] and [BSa1972]. A group is called a Coxeter group (and the pair a Coxeter system), if it admits a presentation of the following form:
where with and is the Coxeter matrix. In other words, is generated by (in the cases we are interested in, only finitely many) elements of order two; and by saying what the orders of the products of two such are, we have given a complete set of relations. The relations (1.1) are clearly equivalent to the following:
The Artin group (named this way and studied in [BSa1972]) belonging to (or to denoted by is the group we get by forgetting the relation (1.2)a. So it has a presentation:
The pair is called an Artin system. Note that we can suppose that for and from now on we do so. We have also the Coxeter graph. This is a graph, having for each a vertex; and two vertices are joint by an edge if and if then the number is written along the edge.
An Artin group or Coxeter matrix or graph is called of finite type if the associated Coxeter group is finite. The Coxeter graphs (and hence the groups) of finite type are classified ([Bou1968]).
(1.4) Example. Braid groups.
Artin groups arise as fundamental croups of certain regular orbit spaces associated with the Coxeter groups. Let us recall the classical example. The permutation group on elements, is generated by the interchanges And it is well known that is a Coxeter system. Then if and if So the Coxeter graph looks like:
This group acts on by permuting coordinates. This action is free, when we restrict it to points, where all the coordinates are different. Dividing out the action we get the space of unordered of different complex numbers. The fundamental group of this space (choosing for as basepoint) can easily be visualized as the so-called braid group on
Intuitively it is clear that is generated by where is a braid of the form:
Also we obviously have the following relations:
Artin showed that together with this relations forms a presentation of So in fact he proved, that the braid group is the Artin group associated with the permutation group. Brieskorn [Bri1971] proved a similar theorem for all finite Coxeter groups. The case of affine Coxeter groups is considered by Nguyen Viet Dung [Ngu1983]. In this chapter we want to state and prove a theorem giving a geometric interpretation of Artin groups as fundamental groups of certain regular orbit spaces of Coxeter groups, generalising these results. The natural context for this theorem is the theory of "linear Coxeter groups" of Vinberg, which we will discuss next.
In this paragraph we want to summarise some of the results of Vinberg [Vin1971]. He considers a set of reflections in the faces of an open polyhedral cone (cf.(V1)), such that the images of the cone under the elements of the group generated by the reflections, are mutually disjoint (cf.(V2)). To be more exact:
(2.1) Definition. Let be a finite dimensional real vector space. A pair is called a Vinberg pair if is a reflection in a hyperplane for and satisfying the following two conditions:
|(V1)||The polyhedral cone property:|
|(V2)||The Tits property: for all where is the group generated by|
(2.2) Theorem (Vinberg). Let be a Vinberg pair, then
|(a)||The set is a convex cone and acts properly discontinuously on the interior|
|(b)||If then the stabiliser of is the group generated by and iff this group is finite.|
|(c)||The pair is a Coxeter system.|
See Vinberg [Vin1971] theorem 2, page 1092.
(2.3) Definition. A Coxeter system (respectively Coxeter group) as obtained from a Vinberg pair by (2.2)(c) is called a linear (or geometric) Coxeter system (respectively group). We often will suppose, that a Vinberg pair is fixed.
(2.4) Corollary. The interior is exactly the set of points in with finite stabiliser.
(2.5) Corollary. We have iff is finite.
(2.6) Corollary. Let denote the set of reflection hyperplanes of reflections of (the "mirrors" of Then is locally finite on is a hyperplane configuration (cf.(I.2.6)).
|Proof of (2.6).|
Since the action of on is properly discontinuous, a point has a neighbourhood of such that the stabiliser of So if then the reflection in is in which is finite.
Vinberg also gives a complete theory how to construct Vinberg pairs (and so linear Coxeter groups). One consequence of this is the existence of at least one linear Coxeter group isomorphic to a given abstract Coxeter group. So the geometric interpretation of Artin groups given in the next paragraph applies to all of them. Another consequence is an easy way to introduce the so-called extended Coxeter (or Weyl) groups in chapter III. So we briefly describe these results. First a result from linear algebra. Let be a linear space and its dual. For svstems we define the following invariants:
The set will be called the characteristic of the system The space (respectively is a subspace of (respectively the space of linear relations among the columns (respectively rows) of For if then for each we have Let denote the codimension of in and the codimension of in Such a system is called isomorphic to another if there exists an isomorphism from to transforming into and into
(2.8) Proposition. Let be any let and be subspaces of and respectively, and let be a nonnegative integer. Then there exists a system unique up to isomorphism, having as its characteristic. Also:
See Vinberg [Vin1971] prop. 15.
Let be the linear map given by for If is a reflection, then and has to be the eigenvector with eigenvalue so then one has hence the necessity of the first part of condition (C2) below.
Let If say then if then all are zero or at least one is negative. So we have seen why (C2) below is necessary.
(2.10) Proposition Vinberg. Let the system have characteristic Let and be as above. Then is a Vinberg pair if and only if the following conditions are satisfied:
|(C1)||for and if then|
|(C2)||for and or for a positive integer|
Agreeing that if then is exactly the Coxeter matrix of the Coxeter system
See Vinberg [Vin1971] theorem 5. p 1105.
(2.11) Remarks. The polyhedral cone property (2.1)(V1) follows from (C1), (C3) and for
If the polyhedral cone property holds, then (C1) and (C2) is equivalent to the Tits property (2.1)(V2).
Suppose we have a Cartan matrix satisfying (C1) and (C2). By choosing we can always force (C3) to be true. In this case is even a simplicial cone.
Now let be an arbitrary Coxeter matrix. Then the matrix which has by definition entries satisfies (C1) and (C2) (if we agree, that The linear Coxeter group with characteristic is called the Tits (linear) representation of the abstract Coxeter group associated to This linear Coxeter group is reduced (cf. [Vin1971]). Note that so in this case (cf. Bourbaki [Bou1968] Ch.V §4.4 "La representation contraorediante"). So we nave:
(2.12) Corollary. Each abstract Coseter group is isomorphic to a linear Coxeter group.
(2.13) Finally let be a Cartan matrix satisfying (C1) and with integer entries. Then (C2) is automatically satisfied with or as or respectively. Choosing and we get a linear Coxeter group acting on a lattice This leads to the notion of an extended Coxeter group and is connected with the notion of a generaliserd root system. This situation will be treated in chapter III.
After this exposition of the theory of Vinberg we conclude this paragraph by studying the Tits galleries of the hyperplane configuration associated to a linear Coxeter group of a Vinberg pair.
(2.14) Lemma. There is a bijective correspondence between the set of conjugates of the generators in and the set of mirrors of
In general each element is a wall of some chamber. For a point in the cone has, by the locally finiteness, a neighbourhood meeting finitely many facets inside At least one of these has the same dimension as M, hence is a face of some chamber. In this case this implies that is of the form for some and So if we know that for all and we have:
then sets up the desired bijective correspondence. Now (2.15) is equivalent to
To prove ($) note that
So if then On the other hand if then and are both reflections fixing which is the closure of some face of and the stabiliser of this face is the group (by theorem (2.2)(b)). So then
(2.16) Proposition. For each sequence the sequence:
is a Tits gallery and each Tits gallery starting in is obtainable this way from a unique sequence
This Tits gallery is direct if and only if the length of as element of the Coxeter group is equal to
We have where So and the sequence (2.17) is indeed a Tits gallery. Moreover we see that is its sequence of mirrors. According to the definition (see (1.3.21) and (1.3.20)(a)) this Tits gallery is direct iff all the mirrors are different. Lemma (2.14) tells us that this is the case iff all the conjugates are different. By Bourbaki [Bou1968] Ch. IV §1.4 lemme 2 this in turn is equivalent to the condition in the proposition. Let be a Tits gallery and suppose by induction that we have proved, that there is a unique sequence such that is given as in (2.17) for The chambers and are separated by exactly one common wall, so also and say by Note that is unique. Then so
(2.18) Example. If and then factors) has length as is well known.
Let be a linear Coxeter system (2.3) in a space and let and be as in §2. The group acts also properly discontinuously on the domain:
The regular points of the action, which are by definition the points with trivial stabiliser are the points of the space:
Now the action of on is proper and free, so we get a space
which is a complex manifold. This space is called the regular orbit space associated to the linear Coxeter group.
(3.4) Let denote the canonical projection. Choose a point
Let and as base point for For define the curve in from to by:
where is continuous and and
(3.7) Finally let
Now we can state and prove the main theorem of this chapter:
(3.8) Theorem. The fundamental group of a regular orbit space associated to a linear Coxeter system is the Artin group associated to the abstract Coxeter system. More precisely, with the notations above:
The group is generated by and a complete set of relations is:
where and each side has factors.
For the proof we need some more preparation, which is also of general use. First we discuss the effects of a group acting on a general hyperplane system. The assertions are clear from naturality and also exact proofs are not difficult.
(3.10) Discussion. Let be a hyperplane system. Let be a group with the property that for all and we have:
Then there is a functorial action of on and This means that permutes and if (or then (respectively and
Suppose we are given in each chamber a point such that for all and Then the subgroupoid of the fundamental groupoid of the space is defined and the functor (respectively as occuring in thm. (I.2.21) (respectively (I.4.10)) is equivariant under the action of on (respectively and the natural action of on
(3.12) Proposition. If moreover permutes simply transitively, then is a semigroup freely generated by the of the following one step galleries:
for some fixed chamber and all and
If acts transitively on is a category with only one object, hence a semigroup. Now fix If the action is simply transitive, then each one step gallery is in the of a unique one step gallery of the form (3.13). Since is by definition freely generated by the one step galleries the freeness for follows too.
A linear Coxeter group satisfies (3.11) and the condition of (3.12), so we have:
(3.14) Corollary. If is the simple hyperplane system of a linear Coxeter group then is a free semigroup, freely generated by where is the of the one step gallery:
Furthermore a gallery is in the
if and only if all and for
If and has length in the Coxeter group, then the galleries in the (3.15) are direct.
To prove equality of the signs we note that if is a one-step gallery and then For if then for instance The rest of the assertions follow easily from (2.16), (3.10) and (3.12).
|Proof of theorem (3.8).|
First note that the the space is the same as the space as defined in (1.2.9). We have already the point (3.5) and the group satisfies the conditions of the discussion (3.10) and proposition (3.12). So the groupoid is defined and the projection induces a functor From the fact that and the unique path lifting property it follows that is surjective and it is also easy to see that iff for some So we have a natural
According to the discussion.in (3.10) the map is equivariant for the action of on and so we have a map Now is surjective, for is so and is a homomorphism of semigroups because is a functor. From the equivariance it follows that the equivalence relation on "having the same is generated by the of the pairs, which generate the gallery equivalence relation. By remark (1.5.4) for these we can take the cancel relations and the positive flip relations.
If we consider the of the cancel relations (I.5.2) and we see that we get all of them by taking and so they are the of and so we get the pairs:
Now we consider the of the positive flip relations Again we can suppose now formula (I.5.3). If is a plinth of then since there are two walls of say and such that is the support of The group generated by and stabilises so it is finite (theorem (2.2)(b)), hence and it follows from proposition (2.16) that and where with factors) and with factors) are two, hence the two direct (by example (2.18)) Tits galleries from to
On the other hand if and then the set is nonempty as follows from Vinberg [Vin1971], theorem 7. pg 1114. Its is finite since so and is a plinth of
It follows that the of the positive flip relations are:
Summarising we have proved the following: We have a surjective homomorphism from the free semigroup freely generated by to the group and the natural isomorphism sending to and to and the images of two are equal iff they are equivalent by means of the cancel relations (3.17) and the flip relations (3.18). This proves the theorem.
(3.19) Remark. Two words in the elements are minimal representatives of the same element in if and only if they are equivalent by means of the relations (both sides factors). This follows from (I.3.26) and (2.16). It implies that induces a map from to which is injective since the projection is a left inverse. But the map is not a homomorphism!
With the help of the geometrical interpretation of Artin groups we shall prove some of their algebraic properties. For instance, we show in this way that the subgroup generated by a subset of the generators forms an Artin system. And the intersection of two such groups is the group generated by the common generators of the two. These facts were proved for so called Artin groups of large type (which means that for so there are no commution relations by Appel [ASc1983]. First some preparations. Let be a linear Coxeter group with Vinberg pair
(4.1) Proposition. (cf. Bourbaki [Bou1968] Ch.V §4.6) The closure is a fundamental domain for the action of on in the following strong sense: if and such that then
Since we see that if then and meet, hence are contained in So The element can be written as a product of reflections in the hyperplanes separating and and these all contain as we saw. Hence
(4.2) Definition. Let and suppose that is a Vinberg pair for some then we denote the linear Coxeter group, the set of mirrors, the Tits cone, the space of regular points, the regular orbit space and the Artin group by and respectively. We call a subset of facettype (w.r.t. a facet if the support of where the facet is in For instance for the Tits representation all subsets of are of facettype.
(4.3) Remark. In fact for each there is such a (cf. Vinberg [Vin1971] Cor. 3 pg. 1105). This follows from theorem (2.10). We need the fact only in the case that is of facettype and then the proof is simple and will be given (4.6)(i).
(4.4) Definition. Let and let be a left coset of Then we define the star of by where is defined as In particular we define Note that
|(i)||If then and|
The only not obvious fact of (i) and (ii) is
For this suppose
we have by (4.1) that
(iii) is proved analogously as in Bourbaki [Bou1968] Ch. V §4.6 prop. 6(i).
(4.6) Proposition. Suppose is a Vinberg pair and let such that and for Hence Let be of facettype by a facet Set Then:
|(i)||is also a Vinberg pair.|
|(iv)||If is finite, then is open in hence contained in|
(i) For we have so there exists a point such that for But then also so So the polyhedral cone property (2.1)(V1) holds also for the subpair To check the Tits property let where and Let choose such that for all with and for all with Then for all and for so Similar we have that so hence
(ii) To prove let first and then for so Conversely, let Choose then if and if so hence
(iii) For the other inclusion it suffices to prove i.e. if Then even Let (respectively be the chamber in that contain (resp. Let be a sequence of elements of determining a direct Tits gallery from to in the hyperplane configuration This determines an element such that Now for all in particular So hence so
(iv) Set Then always and we prove the converse in the case that is finite (which proves (iv), because if is finite then is open in Let Since if finite, we have so for an Let and Then
(v) Let We can suppose that Since we have with finite and Then is an open neighbourhood of in
(4.7) Remark. If is finite, then is of facettype. (cf. Vinberg [Vin1971] theorem 7. pg. 1114). Hence (v) is true for general As we saw (remark (4.3)) this is also true for (i). Perhaps for (ii) and (iii) too, where has to be replaced by
(4.8) Definition. We have already defined (4.2) the space We also define: for Note that and where
(4.9) Proposition. Let be of facettype. Then there exists a map with properties:
|(i)||The map is a homotopy inverse of the inlusion|
|(ii)||For another we have|
Let Then for Choose a closed ball (in some metric on such that for Then we have For each there is a unique such that Then is a continuous map Extend to a map by where is such that This is well defined by (4.1).
We show that is continuous on Let be a left coset, where such that is finite. Now is continuous on and since it is so on where is a finite union of these, in closed sets, so is continuous on Sets of this type are open by (4.6)(iv) and cover hence is continuous on
Since for each and iff we have for and
From ($) it folows that and have the same stabiliser in Since the stabiliser of in and is also the same. It follows with the help of (II.2.2)(b) that Since He have:
By ($) and ($$) we see that we can extend to a map by From ($) it follows also that is contained in (respectively if (respectively From this we see that (respectively Hence is an homotopy inverse of This proves (i).
To prove (ii) it suffices to see that, if then Now so and since is convex we ahve
Now we come to the study of the abstract Artin groups. Let be an (abstract) Coxeter group.
(4.10) Definition. The kernel of the projection of the associated Artin group on the Coxeter group: is called the coloured Artin group and denoted by
We have an exact sequence Note that in the case of a linear Coxeter group, where serves as a base point of as before.
(4.11) Lemma. Let Then there is a natural injective homomorphism
We may suppose that is a Tits linear Coxeter group (II.2.11).
Then is of facettype, hence by lemma (4.9) is a homotopy equivalence, so has a homotopy left inverse. Hence is injective. Since is natural isomorphic to by the homotopy equivalence we get the desired injective homomorphism
So from now on we can consider as a subgroup of
(4.12) Lemma. Let then
We can again suppose that is a Tits linear group. Consider By proposition (4.5)(ii) we have Define the abbreviations:
Let be the map as exists by (4.9). Now from (4.9)(ii) it follows that So and since by (4.9)(i) we see that Since the other inclusion is obvious, we have that what was to be proved.
(4.13) Theorem. Let be an Artin System.
|(i)||If then the subgroup generated by forms with these generators itself an Artin system. In particular the Artin group can we seen as subgroup of|
Let and be the natural homomorphisms. We know already that is injective. We have a commutative diaqram with exact rows and columns. That is injective follows from the fact that and Cf (4.11) are so and by diagram chasing. This proves the first part or the theorem.
(ii) It is clear that To prove the other inclusion suppose that Then By remark (3.19) we can consider as an element of Then Since we also have
This is an excerpt of a thesis entitled The Homotopy Type of Complex Hyperplane Complements, written by Harm van der Lek in 1983.