Chapter II. Artin groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 7 May 2013

§1. Coxeter and Artin groups

First we recall some definitions and facts from [Bou1968] and [BSa1972]. A group W is called a Coxeter group (and the pair (W,{σ1,,σ}) a Coxeter system), if it admits a presentation of the following form:

(1.1) generators:σ1 ,,σ. relations:(σi,σj)mij=1, i,j{1,,}.

where {mij}i,h{1,,}, with mijN{} and mii=1 is the Coxeter matrix. In other words, W 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:


  1. σi2=1, i=1,,
  2. σiσjσi= σjσiσj (each side mijfactors and ij)

The Artin group (named this way and studied in [BSa1972]) belonging to W (or to {mij}), denoted by AW, is the group we get by forgetting the relation (1.2)a. So it has a presentation:

(1.3) generators:s1,,s. relations: sisjsi= sjsisj (each sidemijfactors andi j)

The pair (AW,{s1,,s}) is called an Artin system. Note that we can suppose that mij2 for ij and from now on we do so. We have also the Coxeter graph. This is a graph, having for each i{1,,} a vertex; and two vertices are joint by an edge if mij3 and if mij4, then the number mij 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 +1 elements, S+1 is generated by the interchanges σi(i-1i), i=1,,. And it is well known that (S+1,{σ1,,σ}) is a Coxeter system. Then mij=2 if |i-j|>1 and mij=3 if |i-j|=1. So the Coxeter graph looks like:


This group acts on +1 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 +1-tuples of different complex numbers. The fundamental group of this space (choosing for (0,1,,)+1 as basepoint) can easily be visualized as the so-called braid group on +1-strings: B+1.

Intuitively it is clear that B+1 is generated by {s1,,s} where si is a braid of the form:

0 i-1 i Figure 13.

Also we obviously have the following relations:

sisj=sjsi if|i-j|2 sisi+1si= si+1sisi+1 i=1,,-1 0 i-1 i i+1 si si+1 si 0 i-1 i i+1 si+1 si si+1 Figure 14.

Artin showed that {s1,,s} together with this relations forms a presentation of B+1. 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.

§2. Linear Coxeter Groups

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 𝕍 be a finite dimensional real vector space. A pair (C,{σ1,,σ}) is called a Vinberg pair if σiGl(𝕍) is a reflection in a hyperplane Mi for i=1,, and CK({M1,,M}), satisfying the following two conditions:

(V1) The polyhedral cone property: C={M1,,M}
(V2) The Tits property: w(C)Cw=id𝕍 for all wW, where WGl(𝕍) is the group generated by {σ1,,σ}.

(2.2) Theorem (Vinberg). Let (C,{σ1,,σ}) be a Vinberg pair, then

(a) The set IwWw(C) is a convex cone and W acts properly discontinuously on the interior I.
(b) If xC, then the stabiliser of x is the group generated by {σi|xMi} and xI iff this group is finite.
(c) The pair (W,{σ1,,σ}) 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 I is exactly the set of points in I with finite stabiliser.

(2.5) Corollary. We have I=𝕍 iff W is finite.

(2.6) Corollary. Let denote the set of reflection hyperplanes of reflections of W (the "mirrors" of W). Then is locally finite on I. (V,I,) is a hyperplane configuration (cf.(I.2.6)).

Proof of (2.6).

Since the action of W on I is properly discontinuous, a point xI has a neighbourhood of U such that w(U)UwWx, the stabiliser of x. So if UM, M, then the reflection in M is in Wx, 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 𝕍 be a linear space and 𝕍* its dual. For svstems v1,,v𝕍, ϕi,,ϕ𝕍*, we define the following invariants:


  1. The Cartan matrix A={aij}i,j=1,,, aij=ϕi(vj);
  2. The space Lv of linear relations among the vi, i.e. the kernel of the map (x1,,x)x1v1++xv𝕍;
  3. The space Lϕ of linear relations among the ϕi;
  4. The defect d=dim(L/L[v1,,v]), where LAnn([ϕ1,,ϕ]).

The set {A,Lv,Lϕ,d} will be called the characteristic of the system {vi,ϕi}. The space Lv (respectively Lϕ) is a subspace of Lc(A) (respectively Lr(A)), the space of linear relations among the columns (respectively rows) of A. For if (x1,,x)Lv, then for each i{1,,} we have x1ai1++ xai= x1ϕi(v1) ++ xϕi(v) =ϕi(x1v1++xv) =ϕi(0)=0. Let dv denote the codimension of Lv in Lc(A), and dϕ the codimension of Lϕ in Lr(A). Such a system is called isomorphic to another { vi𝕍, ϕ1 (𝕍)*, i=1,, } , if there exists an isomorphism from 𝕍 to 𝕍 transforming vi into vi and ϕi into ϕi.

(2.8) Proposition. Let A be any ×-matrix, let Lv and Lϕ be subspaces of Lc(A) and Lr(A) respectively, and let d be a nonnegative integer. Then there exists a system { vi𝕍,ϕi 𝕍*,i=1,, } unique up to isomorphism, having {A,Lv,Lϕ,d} as its characteristic. Also:

(2.9) dim(𝕍)=rank(A)+dv+dϕ+d.


See Vinberg [Vin1971] prop. 15.

Let σi be the linear map given by σi(x)x-ϕi(x)vi for x𝕍. If σi is a reflection, then aii0 and vi has to be the eigenvector with eigenvalue -1, so then one has aii=2, hence the necessity of the first part of condition (C2) below.

Let C{x𝕍|ϕi(x)>0,i=1,,}. If C, say cC, then if x1ϕ1(c) ++ xϕ(c) =0, then all x1,,x are zero or at least one is negative. So we have seen why (C2) below is necessary.

(2.10) Proposition Vinberg. Let the system {vi𝕍,ϕi𝕍*,i=1,,} have characteristic {A,Lv,Lϕ,d}. Let C and σ1,,σ be as above. Then (C,{σ1,,σ}) is a Vinberg pair if and only if the following conditions are satisfied:

(C1) aij0 for ij and if aij=0, then aji=0.
(C2) aii=2 for i=1,, and aijaji4 or aijaji=4cos2(π/mij) for a positive integer mij.
(C3) Lϕ { (x1,,x) Lr(A)|xi 0,i=1,, } ={0}.

Agreeing that mij=, if aijaji4, then {mij}i,j=1,, is exactly the Coxeter matrix of the Coxeter system (W,{σ1,,σ}).


See Vinberg [Vin1971] theorem 5. p 1105.

(2.11) Remarks. The polyhedral cone property (2.1)(V1) follows from (C1), (C3) and aii0 for i=1,,.

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 Lϕ={0} we can always force (C3) to be true. In this case C is even a simplicial cone.

Now let M={mij}i,j=1,, be an arbitrary Coxeter matrix. Then the matrix Cos(M), which has by definition entries -2cos(π/mij) satisfies (C1) and (C2) (if we agree, that π/=0). The linear Coxeter group with characteristic (Cos(M),Lv(Cos(M),0,0)) is called the Tits (linear) representation of the abstract Coxeter group associated to M. This linear Coxeter group is reduced (cf. [Vin1971]). Note that dim(Lv(Cos(M))) =-rank(Cos(M)), so in this case dim(𝕍)=. (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 N={nij}i,j=1,, be a Cartan matrix satisfying (C1) and nii=2 with integer entries. Then (C2) is automatically satisfied with mij=2,3,4,6 or as nijnji=0,1,2,3, or 4 respectively. Choosing Lv=0 and Lϕ=0 we get a linear Coxeter group acting on a lattice Qi=1vi. 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 (𝕍,I,) 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 σi in W and the set of mirrors of W ().


In general each element M is a wall of some chamber. For a point xM in the cone I has, by the locally finiteness, a neighbourhood meeting finitely many facets inside M. At least one of these has the same dimension as M, hence is a face of some chamber. In this case this implies that M is of the form w(Mi) for some wW and i{1,,}. So if we know that for all w,vW and i,j{1,,} we have:

(2.15) wσiw-1= vσjv-1 w(Mi)= v(Mj)

then wσiw-1 w(Mi) sets up the desired bijective correspondence. Now (2.15) is equivalent to

($) σi=wσjw-1 Mi=w(Mj) forwWand i,j{1,,}

To prove ($) note that

σi is a reflection with reflection hyperplaneMi wσjw-1 is a reflection with reflection hyperplanew(Mj).

So if σi=wσjw-1, then Mi=w(Mj). On the other hand if Mi=w(Mj), then σi and wσjw-1 are both reflections fixing MiC, which is the closure of some face of C and the stabiliser of this face is the group {id𝕍,σi} (by theorem (2.2)(b)). So then wσjw-1=σi.

(2.16) Proposition. For each sequence i1,,ik{1,,} the sequence:

(2.17) C,C1,,Ck with Cr=σi1σi2σir(C) r=1,,k

is a Tits gallery and each Tits gallery starting in C is obtainable this way from a unique sequence i1,,ik.

This Tits gallery is direct if and only if the length of σi1 σi2 σik as element of the Coxeter group is equal to k.


We have (Cr-1,Cr)= (wr(C),wrσir(C)) =wr((C,σir(C))) ={wr(Mir)} where wrσi1σi2σir-1. So |(Cr-1,Cr)|=1 and the sequence (2.17) is indeed a Tits gallery. Moreover we see that Mi1= w1(Mi1), w2(Mi2), , wk(Mik), 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 w1σi1w1-1, , wkσikwk-1 are different. By Bourbaki [Bou1968] Ch. IV §1.4 lemme 2 this in turn is equivalent to the condition in the proposition. Let C,C1,,Ck be a Tits gallery and suppose by induction that we have proved, that there is a unique sequence i1,,ik-1 such that Cr is given as in (2.17) for r=1,,k-1. The chambers Ck-1 and Ck are separated by exactly one common wall, so also C=wk-1-1(Ck-1) and wk-1-1(Ck), say by Mik. Note that ik is unique. Then wk-1-1(Ck) =σik(C), so Ck= wk-1σik(C) =σi1σik(C).

(2.18) Example. If ij and kmij, then σiσjσi (k factors) has length k, as is well known.

§3. The Regular Orbit Space

Let (W,{σ1,,σ}) be a linear Coxeter system (2.3) in a space 𝕍 and let ϕi,C,Mi,I,I and be as in §2. The group W acts also properly discontinuously on the domain:

(3.1) ΩIm-1(I)𝕍

The regular points of the action, which are by definition the points with trivial stabiliser are the points of the space:

(3.2) YΩ- {reflecetion hyperplanes}=Ω- MM.

Now the action of W on Y is proper and free, so we get a space

(3.3) XY/W

which is a complex manifold. This space is called the regular orbit space associated to the linear Coxeter group.

(3.4) Let p:YX denote the canonical projection. Choose a point

(3.5) cC.

Let *^-1c and *p(*^) as base point for X. For i=1,, define the curve 𝒮i in Y from *^ to σi(*^) by:

(3.6) 𝒮i(t) v(t)+ -1 ((1-t)c+tσi(c))

where v:[0,1]𝕍 is continuous and v(1)=v(0)=0 and ϕi(v(?))>0.

(3.7) Finally let si{p𝒮i}π(X,*).

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 π(X,*) is generated by s1,,s and a complete set of relations is:

(3.9) sisjsi= sjsisj

where mij< and each side has mij 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 (𝕍,H,,p) be a hyperplane system. Let WGl(𝕍) be a group with the property that for all wW, M and NpM we have:

(3.11) w(M)and w(N)pw(M) .

Then there is a functorial action of W on Pgal and Gal. This means that W permutes K() and if GPgalAB (or GalAB), then w(G)Pgalw(A)w(B) (respectively Pgalw(A)w(B)) and w(G1G2)=w(G1)w(G2).

Suppose we are given in each chamber CK() a point aC such that w(aC)=aw(C) for all wW and CK(). Then the subgroupoid Π of the fundamental groupoid of the space Y(𝕍,H,,p) is defined and the functor φ:Pgal (respectively Gal)Π, as occuring in thm. (I.2.21) (respectively (I.4.10)) is equivariant under the action of W on Pgal (respectively Gal) and the natural action of W on Π.

(3.12) Proposition. If moreover W permutes K() simply transitively, then Gal/W is a semigroup freely generated by the W-orbits of the following one step galleries:

(3.13) CUspM(C)

for some fixed chamber CK() and all MC and UK(pM).


If W acts transitively on K(), Gal/W is a category with only one object, hence a semigroup. Now fix CK(). If the action is simply transitive, then each one step gallery AUB is in the W-orbit of a unique one step gallery of the form (3.13). Since Gal is by definition freely generated by the one step galleries the freeness for Gal/W follows too.

A linear Coxeter group satisfies (3.11) and the condition of (3.12), so we have:

(3.14) Corollary. If (𝕍,I,) is the simple hyperplane system of a linear Coxeter group W, then Gal/W is a free semigroup, freely generated by S1+,, S+, S1-,, S- where Siε is the W-orbit of the one step gallery:

Cεσi(C) ε{+,-}, i{1,,}

Furthermore a gallery Cε1 C1ε2 εkCk is in the W-orbit

(3.15) Si1δ1 Si2δ2 Simδm Gal/W

if and only if k=m, εi=δi all i-1,,k and for r=1,,k

Cr= σi1 σi2 σir(C).

If δ1==δm=+ and σi1σim has length m in the Coxeter group, then the galleries in the W-orbit (3.15) are direct.


To prove equality of the signs we note that if AεB is a one-step gallery and wW then w(AεB)= w(A)εw(B). For if (A,B)={M}, then for instance (ε=+) w(DM(A))= Dw(M)(w(A)). 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 Y is the same as the space Y(𝕍,I,) as defined in (1.2.9). We have already the point cC (3.5) and the group W satisfies the conditions of the discussion (3.10) and proposition (3.12). So the groupoid Π is defined and the projection p:YX induces a functor p*:Ππ(X,*). From the fact that p-1(*)={-1w(c)|wW} and the unique path lifting property it follows that p* is surjective and it is also easy to see that p*(λ)=p*(μ) iff λ=w(μ) for some wW. So we have a natural

(3.16) isomorphism: p*/W:Π/W π(X,*)

According to the (3.10) the map φ:GalΠ is equivariant for the action of W on Gal and Π, so we have a map φ/W:Gal/WΠ/W. Now φ/W is surjective, for φ is so and φ/W is a homomorphism of semigroups because φ is a functor. From the equivariance it follows that the equivalence relation on Gal/W "having the same φ/W-image" is generated by the W-orbits 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 W-orbits of the cancel relations (I.5.2) (A+B-A,A) and (A-B+A,A) we see that we get all of them by taking A=C and B=σi(C) i=1,,, so they are the W-orbits of (C+σi(C)-C,C) and (C-σi(C)+C,C) so we get the pairs:

(3.17) (Si+Si-,) and (Si-Si+,) i=1,,

Now we consider the W-orbits of the positive flip relations Again we can suppose C0=D0=C now formula (I.5.3). If P is a plinth of C, then since dim(P)=2 there are two walls of C say Mi and Mj such that MiMj is the support of P. The group generated by σi and σj stabilises PI so it is finite (theorem (2.2)(b)), hence mij< and it follows from proposition (2.16) that C0,C1,..,Cmij and D0,D1,,Dmij, where Crwr(C) with wrσiσjσi (r factors) and Drwr(C) with wrσjσiσj (r factors) are two, hence the two direct (by example (2.18)) Tits galleries from C to spP(C).

On the other hand if i,j{1,,} and mij<, then the set P{xW|ϕr(x)>0ifri,jandϕi(x)=ϕj(x)=0}C is nonempty as follows from Vinberg [Vin1971], theorem 7. pg 1114. Its W-stabiliser is finite since mij<, so PI and P is a plinth of C.

It follows that the W-orbits of the positive flip relations are:
(3.18) ( Si+Sj+Si+, Sj+Si+Sj+ ) where i,j{1,,}, ij and mij< and both elements of (3.18) have mij factors. It is easily checked that under the map φ/W followed by the isomorphism p*/W (3.16) the element Si+ is mapped on the element siπ(X,*) as defined earlier (3.7).

Summarising we have proved the following: We have a surjective homomorphism from the free semigroup Gal/W freely generated by S1+,, S+, S1-,, S- to the group π(X,*) (φ/W and the natural isomorphism p*/W), sending Si+ to si and Si- to si-1 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 σi are minimal representatives of the same element in W if and only if they are equivalent by means of the relations σiσjσi= σjσiσj (both sides mij< factors). This follows from (I.3.26) and (2.16). It implies that σisi induces a map from W to AW, which is injective since the projection AWW is a left inverse. But the map is not a homomorphism!

§4. Applications to Abstract Artin groups

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 mij3 for ij, so there are no commution relations mij=2) by Appel [ASc1983]. First some preparations. Let W be a linear Coxeter group with Vinberg pair (C,{σ1,,σ}).

(4.1) Proposition. (cf. Bourbaki [Bou1968] Ch.V §4.6) The closure C is a fundamental domain for the action of W on I in the following strong sense: if xC and wW such that w(x)C, then w(x)=x.


Since w(x)Cw(C), we see that if w(x)M, then C and w(C) meet, hence are contained in DM(w(x)). So M(C,w(C)). The element w can be written as a product of reflections in the hyperplanes separating C and w(C) and these all contain w(x) as we saw. Hence w(x)=x.

(4.2) Definition. Let J{1,,} and suppose that (CJ,{σj|jJ}) is a Vinberg pair for some CJ, 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 WJ, J, IJ, YJ, XJ and AWJ respectively. We call a subset J{1,,} of facettype (w.r.t. a facet F), if jJMj=LF, the support of F, where the facet F is in C. For instance for the Tits representation all subsets of {1,,} are of facettype.

(4.3) Remark. In fact for each J{1,,} there is such a CJ (cf. Vinberg [Vin1971] Cor. 3 pg. 1105). This follows from theorem (2.10). We need the fact only in the case that J is of facettype and then the proof is simple and will be given (4.6)(i).

(4.4) Definition. Let J{1,,} and let H be a left coset of WJ. Then we define the star of H by Star(H)wHw(CJ), where CJ is defined as CJ { xC|x MiiJ } . In particular we define St(J)Star(WJ). Note that Star(w.WJ)=w(St(J)).

(4.5) Proposition.

(i) If JJ, then CJCJ and St(J)St(J).
(ii) CJ1CJ2=CJ1J2 and St(J1)St(J2)=St(J1J2), Ji{1,,}.
(iii) Star(H) is convex.


The only not obvious fact of (i) and (ii) is St(J1)St(J2)St(J1J2). For this suppose xw1(CJ1) w2(CJ2) w1(C) w2(C), with wiWJi. Since w1-1(x) C w1-1w2(C) we have by (4.1) that
($) w1-1w2 w1-1(x) =w1-1(x), hence w1-1(x)x w2-1(x) and this is an element of CJ1CJ2= CJ1J2. From ($) it follows also that w1-1w2 WJ1WJ2= WJ1J2 (The last equality is well known for Coxeter groups). It follows that w1WJ1J2 and xw1(CJ1J2)St(J1J2).

(iii) is proved analogously as in Bourbaki [Bou1968] Ch. V §4.6 prop. 6(i).

(4.6) Proposition. Suppose (C,{σ1,,σ}) is a Vinberg pair and let ϕ1,,ϕ𝕍* such that Mi=ϕi-1(0) and ϕi(C)+ for i=1,,. Hence C={x𝕍|ϕi(x)>0,i=1,,}. Let J{1,,} be of facettype by a facet F. Set CJ={x𝕍|ϕi(x)>0,jJ}. Then:

(i) (CJ,{σj|jJ}) is also a Vinberg pair.
(ii) CJ=C+LF (Hence also CJ=C+LF).
(iii) IJ=I+LF.
(iv) If WJ is finite, then St(J) is open in 𝕍, hence contained in I.
(v) St(J)I is open.


(i) For jJ we have MjC so there exists a point xCMj such that xMi for ij. But then also xCjMj, so MjCJ. So the polyhedral cone property (2.1)(V1) holds also for the subpair (CJ,{σj|jJ}). To check the Tits property let y=w(x)CJw(CJ), where wWJ and xCJ. Let fF, choose ε>0 such that ε<ϕi(f)-ϕi(x) for all iJ with ϕi(x)<0 and ε<ϕi(f)-ϕi(w(x)) for all iJ with ϕi(w(x))<0. Then ϕi(f+εx)= ϕi(f)+ εϕi(x)>0 for all iJ and ϕj(f+εx)= εϕj(x)>0 for jJ, so f+εxC. Similar we have that w(f+εx)=f+εw(x)C, so w(f+εx)Cw(C), hence w=id𝕍.

(ii) To prove CJ=C+LF let first xC and yLF, then ϕj(x+y)= ϕj(x)>0 for jJ so C+LFCJ. Conversely, let xCJ. Choose λ>maxiJ(-ϕi(x)ϕi(f)), then ϕi(x+λf)=ϕj(x)+λϕi(f)>0 if iJ and ϕj(x+λf)=ϕj(x)>0 if jJ, so x+λfC, hence xC+LF.

(iii) IJ=wWJw(CJ) wWJ (w(C)+LF) I+LF. For the other inclusion it suffices to prove IIJ i.e. w(C)IJ if wW. Then w(C)I even w(C)K(,I). Let C (respectively C) be the chamber in K(J,I) that contain C (resp. w(C)). Let M1,,Mk be a sequence of elements of J determining a direct Tits gallery from C to C in the hyperplane configuration (V,I,J). This determines an element wWJ such that (w)-1 (w(C)) C. Now CDM(C) for all MJ, in particular CCJ. So (w)-1 (w(C)) CJ, hence w(C)w(CJ), so w(C)w(CJ)IJ.

C C C w(C) Figure 15.

(iv) Set A { xV|ϕi (w(x))>0 for alliJand wWJ } . Then always St(J)A and we prove the converse in the case that WJ is finite (which proves (iv), because if WJ is finite then A is open in 𝕍). Let xA. Since WJ if finite, we have x𝕍=IJ=wWJw(CJ), so x=w(y) for an yCJ=C+LF. Let y=y+y, yC and yLF. Then

ϕi(y) = ϕi(w-1(x)) >0ifiJfor xA, ϕj(y) = ϕj(y)+ ϕj(y) =ϕj(y)0 ifjJ.

So yCJ, hence xSt(J).

(v) Let xSt(J)I. We can suppose that xCJ. Since xI, we have xCJ with WJ finite and JJ. Then St(J) is an open neighbourhood of x in St(J)I.

(4.7) Remark. If WJ is finite, then J is of facettype. (cf. Vinberg [Vin1971] theorem 7. pg. 1114). Hence (v) is true for general J. As we saw (remark (4.3)) this is also true for (i). Perhaps for (ii) and (iii) too, where LF has to be replaced by jJMj.

(4.8) Definition. We have already defined (4.2) the space YJY(𝕍,IJ,F). We also define: Y(J)Y(𝕍,ISt(J),) for J{1,,}. Note that Y(J)YYJ and Y(J)=Im-1(St(J)), where Im:YH.

(4.9) Proposition. Let J{1,,} be of facettype. Then there exists a map r:YJY(J) with properties:

(i) The map r is a homotopy inverse of the inlusion i:Y(J)YJ.
(ii) For another J{1,,} we have r(Y(J))Y(J).


Let fF. Then ϕj(f)>0 for jJ. Choose a closed ball B (in some metric on 𝕍), such that ϕj(B)+ for jJ. Then we have BCJCJ. For each yCJ there is a unique r(y)BCJ such that [f,r(y)]=B[f,y]. Then r is a continuous map CJCJ. Extend to a map r:IJSt(J) by r(y)w-1(r(w(y))), where wWJ is such that w(y)CJ. This is well defined by (4.1).

We show that r is continuous on IJ. Let HwWJ be a left coset, where JJ, such that WJ is finite. Now r is continuous on CJ and since CJ CJCJ, it is so on w.w(CJ), where wWJ. Star(H) is a finite union of these, in Star(H) closed sets, so r is continuous on Star(H). Sets of this type Star(H) are open by (4.6)(iv) and cover IJ, hence r is continuous on IJ.

Since fM for each MJ and r(y)f iff yf, we have for yIJ, yf and MJ:

($) yM [y,r(y)]M [y,r(y)]M r(y)M.

From ($) it folows that y and r(y) have the same stabiliser in WJ. Since r(y)St(J), the stabiliser of r(y) in W and WJ is also the same. It follows with the help of (II.2.2)(b) that r(IJ)IJ. Since r(IJ)St(J)I, He have:

($$) r(IJ) St(J)I.

By ($) and ($$) we see that we can extend r to a map r:YJY(J) by r(x+-1y) x+-1r(y). From ($) it follows also that [z,r(z)] is contained in WJ (respectively Y(J)) if zYJ (respectively zY(J)). From this we see that iridYJ (respectively riidY(J)). Hence r is an homotopy inverse of i. This proves (i).

To prove (ii) it suffices to see that, if ySt(J), then r(y)St(J). Now fC, so fSt(J) and since St(J) is convex we ahve (f,y]St(J).

Now we come to the study of the abstract Artin groups. Let W be an (abstract) Coxeter group.

(4.10) Definition. The kernel of the projection of the associated Artin group on the Coxeter group: AWW is called the coloured Artin group and denoted by CAW.

We have an exact sequence 1CAWAWW1. Note that CAW=π(Y,*^) in the case of a linear Coxeter group, where *^-1c serves as a base point of Y as before.

(4.11) Lemma. Let J{1,,}. Then there is a natural injective homomorphism Cf:CAWJCAW.


We may suppose that W is a Tits linear Coxeter group (II.2.11).

Then J is of facettype, hence by lemma (4.9) Y(J)YJ is a homotopy equivalence, so Y(J)Y has a homotopy left inverse. Hence π(Y(J),*^)π(Y,*^) is injective. Since π(Y(J),*^) is natural isomorphic to π(YJ,*^)=CAWJ by the homotopy equivalence Y(J)YJ, we get the desired injective homomorphism Cf:CAWJAWJ.

So from now on we can consider CAWJ as a subgroup of CAW.

(4.12) Lemma. Let J1,J2 {1,,}, then CAWJ1 CAWJ2= CAWJ1J2


We can again suppose that W is a Tits linear group. Consider Im:YI. By proposition (4.5)(ii) we have Y(J1) Y(J2)= Im-1(St(J1)I) Im-1(St(J2)I)= Im-1 ( St(J1) St(J2)I ) = Im-1 ( St(J1J2) I ) Y(J1J2). Define the abbreviations:

G12 π ( Y(J1) Y(J2), *^ ) =π (Y(J1J2),*^) CAWJ1J2, Gi π(Y(Ji),*^) =CAWJi, G (Y,*^)=CAW

Let r be the map YJ1Y(J1) as exists by (4.9). Now from (4.9)(ii) it follows that r(Y(J2)) Y(J2) Y(J1)= Y(J1J2). So r*(G2)G12 and since by (4.9)(i) r*|G1=idG1, we see that G1G2=r*(G1G2)G12. Since the other inclusion is obvious, we have that G1G2=G12, what was to be proved.

G1 G12 G1G2 G G2 r*

(4.13) Theorem. Let (AW,{s1,,s}) be an Artin System.

(i) If J{1,,}, then the subgroup generated by {sj|jJ} forms with these generators itself an Artin system. In particular the Artin group AWJ can we seen as subgroup of AW.
(ii) Let J1,J2{1,,}. Then AWJ1 AWJ2= AWJ1J2.


Let f:AWJAW and f:WJW be the natural homomorphisms. We know already that f is injective. We have a commutative diaqram with exact rows and columns. That f is injective follows from the fact that f and Cf (4.11) are so and by diagram chasing. This proves the first part or the theorem.

1 1 1 CAWJ AWJ WJ 1 Cf f f 1CAW AW W1

(ii) It is clear that AWJ1 AWJ2 AWJ1J2. To prove the other inclusion suppose that xAWJ1AWJ2. Then π(x)wWJ1WJ2=WJ1J2. By remark (3.19) we can consider w as an element of AWJ1J2. Then xw-1 CAWJ1 CAWJ2= CAWJ1J2 AWJ1J2. Since wAWJ1J2 we also have xAWJ1J2.

Notes and References

This is an excerpt of a thesis entitled The Homotopy Type of Complex Hyperplane Complements, written by Harm van der Lek in 1983.

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