Chapter III. Extended Artin groups

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

Last update: 8 May 2013

§1. Generalised Root Systems and Extended Coxeter groups

We recall some definitions from Looigenga [Loo1980]. Let 𝕍 be a real vector space of finite dimension. A subset B of 𝕍 together with an injection :B𝕍*, αα, αB is called a root basis if the following conditions are satisfied:

(i) B (respectively B) is a linearly independent subset of 𝕍 (resp. 𝕍*).
(ii) α(α)=2 for all αB.
(iii) β(α) is a nonpositive integer for α,βB, αβ.
(iv) β(α)=0 implies α(β)=0.

We order the elements of B={α1,,α}, =|B|. The Cartan matrix N has entries defined by nijαi(αj) (This deviates from the convention in [Lek1981] and [Loo1980], where nij=αj(αi)).

Two root bases (𝕍1,B1) and (𝕍2,B2) are said to be isomorphic if they are so in the sense of (II.2), i.e. if there exists a linear isomorphism f:𝕍1𝕍2 such that f(B1)=f(B2) and f*(B2)=f*(B1).

It follows from proposition (II.2.10) that for each Cartan matrix N with properties (C1) and (C2) and integer valued entries there exists a root basis with this Cartan matrix and two such are isomorphic if the vector spaces in which they are defined have the same dimension.

It follows from (II.2.9) that we have dim(𝕍)+corank(N) and if the root basis is irreducible (see Looiienga [Loo1980]) , the equality holds.

Since condition (C3) of theorem (II.2.10) is automatically satisfied we know that (C,{σ1,,σ}) is a Vinberg pair, where

(1.1) σi(x) x-αi(x) αiand C { x𝕍| αi(x) >0 }

So if as before W denotes the subgroup of Gl(𝕍) generated by the reflections {σ1,,σ} and IwWw(C) then the conclusions of (II.2.2) to (II.2.6) hold.

But now due to the integrality of the Cartan matrix we have more structure: Let Qi=1αi, the so-called (root) lattice. Since σi(Q)Q, W acts on Q. Let W be the group of affine transformations of 𝕍 generated by W and Q. This group is called an extended Coxeter group.

(1.2) Remark. In the sequel often Q will be seen as a subgroup of a non commutative group. Then it is convenient to write an element qQ as eq. For instance QW. In this case we denote the elments eαi by τi.

Since for wW,qQ and x𝕍 we have:

w·eq(x)=w(q+x) =w(q)+w(x)= ew(q)·w(x)

we see that W is the semidirect product of W and Q. So we have the following two proposition, which are given to compare with the various formulations of the main result of this chapter:

(1.3) Proposition. The group W can be described as follows:
Add to the free product of W and Q the following relations:

σi·eq= eσi(q)· σii{1,,} ,qQ.

(1.4) Proposition. The group W is generated by σ1,,σ, τ1,,τ, where τieαi. A complete set of relations is (i,j{1,,}):

  1. (σiσj)mij=1 for mij<.
  2. τiτj= τjτi.
  3. σiτj= τjτi-nijσi.

(1.5) Proposition. If |W|<, then W is an affine Coxeter group (5.1). Then it acts simply transitively on the chambers w.r.t. the reflection hyperplanes (mirrors) of W.


See Bourbaki [Bou1968] Ch. V §3.

The group W acts on 𝕍* by the rule w(τ)(x) τ(w-1(x)), for x𝕍, wW and τ𝕍*. Let RW·B (respectively RW·B) denote the set of roots (respectively dual roots). This construction is called a generalised root system.

(1.6) Definition. For αR and n we introduce:

M(α,n) { x𝕍| α(x)=n } , Mα M(α,0), U(α,n) { x𝕍|n< α(x)<n+1 } , U(α,+) { x𝕍|α (x)>0 } , { Mα|α R } ,the set of mirrors ofW, pα { M(α,n)| n } , p : Mα pα.

(1.7) Proposition. Let αR, wW and n. Then

i) The quadruple (𝕍,I,,p) is a hyperplane system (I.2.7),
ii) w(M(α,n)) =M(w(α),n),
iii) w(U(α,n))= U(w(α),n),
iv) w(U(α,+))= U(w(α),+),
v) q+M(α,n)= M(α,n+α(q)),
vi) q+U(α,n)= U(α,n+α(q)).


Straightforward. For instance vi): xq+U(α,n) x-qU(α,n) n<α(x-q)<n+1 n+α(q)< α(x)<n+ α(q)+1.

§2. The Regular Orbit Space and its Fundamental group

In this chapter the symbols Y and X have a different, although similar meaning as in chapter II. Let have as in §1 a generalised root system and the associated Coxeter (Weyl) group W. As in (II.3) the group W acts properly discontinuously on the domain ΩIm-1(I). Also W acts properly discontinuously on Ω, where the action of Q is given by x+-1yx+q+-1y for qQ, i.e. Q affects the real plane only.

In Ω the points in regular orbits or the regular points, ??? are by definition the points with trivial stabiliser (of the W-action ???) are the points of the space

(2.1) YΩ- {reflection hyperplanes ofW} .

This follows from:

(2.2) Lemma. The ??? of non regular points in Ω is the union of the reflection hyperplanes of W. The reflection hyperplanes are the sets of the form ??? for some αR and n.


See Looijenga [Loo1980] lemma 2.17.

Now we define the regular orbit space associated to the extended Coxeter group (or in fact to the generalised root system) by:

(2.3) XY/Wand let p:YXdenote the projection.

In the fundamental chamber C (1.1) we choose a point c. Now -1cY (from (2.2)) and we take *p(-1c) as a base point for X.

(2.4) Now we define elements s1,,s, t1,,t π(X,*). For i=1,, define:

𝒮i: [0,1] Yby t δ(t)αi+ -1 ( (1-t)c+ tσi(c) )

where δ:[0,1][0,12) is continuous and δ(12)0, e.g. δ(t)=14sin(πt), see figure 16.

𝒯i: [0,1] Yby t tαi+-1c.

Now p𝒮i and p𝒯i are closed curves (loops) in X and they represent by definition the classes si and ti respectively.

12 0 1 12 Figure 16.

Now we can state the main theorem of this chapter:

(2.5) Theorem. The fundamental group π(X,*) of the regular orbit space X of an extended Coxeter group W is generated by two subgroups isomorphic to AW and Q respectively. In turn these groups are generated by {s1,,s} and {t1,,t} respectively, where si and ti are as above.

Furthermore the following relations hold:

(2.6) si(n). eq=eσi(q) ·si(n+αi(q))

for i=1,,, qQ and n, where

(2.7) si(2m) timsi timand si(2m-1) timsi-1 timfor m.

This is a complete set of relations.

(2.8) Remark. The relations (2.6) are called push relations, because they tell how to "push" eq through si(n). This "pushing" seems to be from the right to the left, however we also have:

(2.9) eq·si(n)= si(n+αi(q)) ·eσi(q).

Proof of (2.9).

By (2.6) the right hand side of (2.9) equals eσi(σi(q)) · si ( n+αi(q)+ αi(σi(q)) ) and this equals eq·si(n), for αi(q+σi(q))= αi(2q-αi(q)αi) =2αi(q)- αi(q)αi(αi) =0.

We will prove theorem (2.5) in §3 and §4. Now we first illustrate the elements si(n) and the push relations:

12U(αi,n) αi Mαi C c real part ofsi(n) imaginary part ofsi(n) Figure 17.

Figure 17 depicts a curve in Y, which is the lift of a typical representative of si(n). The imaginary part moves from c to σi(c). The real part starts and ends in the origin, but when the imaginary part is on Mαi, the real part has to be in the layer U(αi,n) { xV|n< ai(x)< n+1 } .

q+U(αi,n)=U(αi,n+αi(q)) } } } n αi(q) n+αi(q) 0 q Figure 18.

Figure 18 depicts the push relations si(n)· eq=eσi(q) ·si(n+αi(q)). A lift of a typical representative of si(n)·eq can be described as follows: First the real part moves from the origin to q. Then the curve described above takes place, the real part being shifted over q. So now, if the imaginary part is on Mαi, the real part is in the layer q+U(αi,n) =U(αi,n+αi(q)). Therefore intuitively we can see that we can also start si(n+αi(q)). The element eσi(q) is then necessary to take care, that the real part ends in the correct point, namely q.

(2.10) Corollary (cf. prop. (1.3)). The group π(X,*) can be described as follows: Add to the free product of AW and Q the push relations (2.6).

As a second reformulation of the main result (2.5) we give an explicit presentation of π(X,*).

(2.11) Corollary (cf. prop. (1.4)). The group π(X,*) is generated by the elements s1,,s,t1,,t, a complete set of relations is:

1b. sisjsi= sjsisj ij,mij<, mij factors both sides.
2. titj=tjti
3b. sitj= tjtirsi ti-r nij even and nij=-2r
3c. sitj= tjtir+1 si-1 ti-r nij odd and nij=-2r-1

where i,j{1,,}.

In fact these corollaries are equivalent to the theorem.

Proof of (2.5) (2.11).

Relations 3b. and 3c. are special cases of the push relations (2.6). Namely, if n=0 and q=αj then:

sitj=ti(0) ·eαj= eσi(αj) ·si(αi(αj))

Now σi(αj)= αj-αi(αj)αi =αj-nijαi, so eσi(αj)= tjti-nij. If nijαi(αj) is even, say -nij=2r, then

sitj=tj ti2rti-r siti-r= tjtirsi ti-r.

And if nij is odd, say -nij=2r+1, then

sitj=tj ti2r+1 ti-r si-1 ti-r=tj tir+1 si-1ti-r .

Conversely it is easy to see that the push relations (2.6) follow from these special cases and the commutation relations 2.

Relations 3b. and 3c. say how one can "push tj through si". But one cannot push ti through si, because for i=j we have -nij=-2, so r=-1 and 3b. becomes siti=siti. This in contrast to W where the relation 3a. σiτj=τi-1σj is valid.

In the special case that nij{0,-1} for all ij, we get a complete set of relations:

(2.12) sisj = sjsi }for nij=0. sitj = tjsi titj = tjti for alli,j {1,,}. sisjsi = sjsisj }for nij=-1. sitjsi = titj

This was conjectured by E. Looijenga in 1978 in a letter to C.T.C. Wall.

It is interesting to see how the presentation of (2.11), similar to what we saw with Coxeter and Artin groups, can be obtained by starting with a presentation of W, changing it in a suitable equivalent presentation and then deleting some relations. In fact this describes precisely the homomorphism π:π(X,*)W.

A presentation of W (1.4): Another presentation of W:
generators: σ1,,σ, τ1,,τ generators: σ1,,σ, τ1,,τ
relations: relations:
1. (σiσj)mij =1
1a. σi2=1
1b. σiσjσi= σjσiσj, ij,
mij< mij factors
2. τiτj=τjτi
2. τiτj=τjτi
3. σiτj= τj τi-nij σi
3a. (i=j) σiτi= τi-1σi
3b. (-nij=2r) σiτj= τjτjr σiτi-r
3c. (-nij=2r+1) σiτj= τjτjr+1 σi-1 σiτi-r

And we see that we get the presentation of corollary (2.11) by deleting the relations:

1a. σi2=1 (W changes into AW)
3a. σiτi= τi-1σj (pushing τi through σi is no longer allowed)

(2.13) Definition. We call the abstract group defined by the presentation (2.11) the extended Artin group AW, associated to the extended Coxeter group W. The number is called the rank of the extended Artin group. Note that it isn't an extension in the usual sense.

So finally we can rephrase the result as:

(2.14) Theorem. The fundamental group of the regular orbit space associated to an extended Coxeter group is the extended Artin group associated to the extended Coxeter group.

These results were announced in van der Lek [Lek1981].

§3. A presentation of π-1(W)

Let π:π(X,*)W be the canonical surjective homomorphism. In this paragraph we derive a presentation of the subgroup it π-1(W) of π(X,*), as a first step in the proof of the main theorem (2.5) and because it is interesting in itself.

The space Y equals Y(𝕍,I,,p), where and p are as defined in (1.6). Recall that below (2.3) we have fixed a point cC. For each wW we define a aw(C)w(c)w(C), so in each chamber a point is given. Now we apply the results of the discussion (II.3.10). So we know that the functor φ:GalΠ gives us a surjective homomorphism of semigroups φ/W:Gal/WΠ/W.

The cover Y/WX induces an injection π(Y/W)π(X) of fundamental groups and the image is exactly π-1(W). Hence π-1(W) is isomorphic to π(Y/W). Now π(Y/W) is canonically isomorphic to Π/W, as can be seen in the same way as (II.3.16). Let ν be the surjective homomorphism obtained by φ/W and this last isomorphism:

(3.1) Gal/W ν π-1(W) π(X,*) φ/W Π/W

Furthermore we saw in (II.3.12) that Gal/W is a free semigroup freely generated by the W-orbits of the one step galleries of the form (II.3.13). In the present situation these one step galleries are of the form:

(3.2) CU(αi,n) σi(C)where i{1,,}and n.

We denote the W-orbit of (3.2) by Si(n).

Now we study the W-orbits of the cancel and flip relations.

(3.3) Lemma. The W-orbits of the cancel relations are:

( Si(n) Si(-n-1), ) fori=1,, andn.


A typical cancel relation is of the form:

($) ( CU(αi,n) σi(C)U(αi,n) C,C )

Since σi(U(αi,n))= U(-αi,n)= { xV|n< -αi(x)<n+1 } = { x𝕍|-n-1< αi(x)<-n } = U(αi,-n-1), the W-orbit of the left element of ($) is Si(n) Si(-n-1) .

(3.4) Definition. If (G,G)Gal×Gal is a flip relation, then we call G flippable and G its flipped form. We use the same terminology for W-orbits.

We collect some properties needed in the sequel. Properties (iii), (iv) and (v) are used in §4.

(3.5) Proposition.

(i) A gallery G= CU1 C1U2 Um Cm is flippable if and only if there are i,j{1,,} with m=mij (so mij<) ij and the W-orbit of G is of the form:
(3.6) g=Si(n1) Sj(n2) Si(n3) (mfactors)for some n1,,nm andU U1Um
(ii) In this case the flipped form of the W-orbit g is:
(3.7) Sj(nm) Si(nm-1) Sj(nm-2) (mfactors)
(iii) If n1=n2==nm=0 and g of the form (3.6), then 0U. In particular U, hence G and g are flippable.
(iv) If the set U has the property that 0U, then nr{0,-1} for r=1,,m.
(v) If G is flippable, then there is a qQ such that 0q+U.


(i) If G is flippable then g has the form (3.6). This follows in the same way as (II.3.18) is proved and the fact that U is part of the definition of flippable. The converse is obvious.

(ii) From the fact that g is the W-orbit of G we know that

(£) Ur=U (α(r),nr) r=1,,mwhere α(r)is some dual root of R{i,j}.

(In fact we have α(1)=αi, α(2)=σi(αj), α(3)=σiσj(αi), etc.) Suppose the flipped form of g is Sj(nm) Si(nm-1) Sj(nm-2) (m factors). We have then that:

(££) Ur=U ( α(r), nr ) for someα (r)R{i,j}

(In fact we have α(m)= αj, α(m-1)= σj(αi), α(m-2)= σjσi(αj), etc.) To be able to conclude from (£) and (££) that nr=nr we have to rule out the possibility that α(r)=-α(r). For this consider the galleries of the simple hyperplane system (𝕍,I,). We know that the signed gallery

C+ C1+ C2+ Cm= CU1 C1U2 C2Um Cm

is flippable and its flipped form is

C+ D1+ D2+ Dm= CUm D1Um-1 D2U1 Dm

where Ur= U(α(r),+) =U(α(r,+)) (definition (1.6)). It follows that α(r) and α(r) are the same (positive) dual roots.

(iii) We know that UU1Um is nonempty so pick an xU. Then for λ>0 small enough we have α(r) (λx)<1 for r=1,,m, so λx r=1m U(α(r),0) U0, hence (iii).

(iv) Is obvious.

(v) If G is flippable, let i,j{1,,} be as in (i) and U0 as in the proof of (iii). Note that 0U0. Set J{i,j}. Then WJ is finite, so WJ is an affine group, acting transitively on the chambers w.r.t. its mirrors (see (1.5)). Since (C,Cm)=J, U is such a chamber and so is U0. So there exists a unique w·eqWJ, wWJ, qQJ such that w·eq(U)=U0. Then 0w·eq(U) and because w(0)=0 we also have 0q+U.

(3.8) Let i,j{1,,} and mmij<. We want to determine explicitly the set Nijm of elements (n1,,nm) such that (3.6) is flippable. Since Nji= { (n1,,nm) m| (nm,,n1) Nij } by (3.5)(ii) we can restrict ourselves to nijnji, so, since nijnji<4, there are four cases: (nij,nji)= (0,0), (-1,-1), (-1,-2) and (-1,-3) respectively. The positive dual roots of α(r)Rij, r=1,,m are linear combinations x(r)αi+ y(r)αj with integer coefficients x(r) and y(r) given by the following table:

CASE1.2.3.4. m2346 (nij,nji) (0,0) (-1,-1) (-1,-2) (-1,-3) r 12 123 1234 123456 x(r) 10 110 1210 132310 y(r) 01 011 0111 011211

As is well known. Let us recall one example how these coefficients can be calculated:

Case 3. (nij,nji)=(-1,-2), then

α(2)=σi (αj)= αj-αj (αi)αi =αj-nji αi=2αi +αj.

Let U be as in (3.5)(i). If zU, then the numbers aαi(z) and bαj(z) satisfy, since α(r)(z)= x(r)a+ y(r)b and zUr=U(α(r),nr), the following:

($) nr<x(r)a+ y(r)b<nr+1 r=1,,m.

Conversely, since αi and αj are linearly independent the existence of numbers a and b such that ($) hold, guarantees that U. Now we look for necessary and sufficient conditions on (n1,,nm) for the existence of a and b with ($) in each special case:

Case 1. nij=nji=0, m=2.

n1<a<n1+1 No conditions on(n1,n2) n2<b<n2+1 soNij=2

Case 2. nij=nji=-1, m=3.

(1) n1<a<n1+1
(2) n2<a+b<n2+1
(3) n3<b<n3+1

(1) and (3) imply n1+n3<a+b<n1+n3+2, so with (2) we see that n2=n1+n3 or n2=n1+n3+1, so n1-n2+n3{0,-1}. And conversely if the latter is satisfied, there are a,b such that (1), (2) and (3) hold. So Nij= { (n1,n2,n3) 3|n1- n2+n3{0,-1} } .

0 -1 n2+1 n2 n1 n1+1 Figure 19.

Case 3. nij=-1 and nji=-2, m=4.

(1) n1<a<n1+1
(2) n2<2a+b<n2+1
(3) n3<a+b<n3+1
(4) n4<b<n4+1

From (1), (3) and (4) it follows that n1-n3+n4{0,-1} and from (1), (2) and (3) it follows that n1-n2+n3{0,-1}. So (n1-n3+n4,n1-n2+n3){0,-1}2 and conversely in each of the four possibilities, there are a,b satisfying (1)-(4). The point (a,b) can be found in the region indicated in figure 20. Hence Nij= { (n1,,n4) 4| (n1-n3+n4,n1-n2+n3) {0,-1}2 } .

00 0-1 -10 -1-1 n4+1 n4 n1 n1+1 Figure 20.

Case 4. nij=-1 and nji=-3, m=6.

(1) n1<a<n1+1
(2) n2<3a+b<n2+1
(3) n3<2a+b<n3+1
(4) n4<3a+2b<n4+1
(5) n5<a+b<n5+1
(6) n6<b<n6+1
0000 000-1 00-10 0-100 0-10-1 0-1-10 -100-1 -10-10 -10-1-1 -1-10-1 -1-1-10 -1-1-1-1 n6+1 n6 n1 n1+1 Figure 21.

As in the previous cases it can be derived that k(k1,k2,k3,k4){0,-1}4 where k is given by k ( n1-n5+n6, n1-n3+n5, n1-n2+n3, n2-n4+n6 ) . But not all 16 possibilities occur for we have an extra condition, for instance k1k3k4=k1. In fact it can be checked that we have the follow: There exist a,b satisfying (1)-(6) if and only if kA, where A { (k1,k2,k3,k4) {0,-1}4| k1k3k4=k1 } . The 12 open regions in the open square (n1,n1+1)×(n6,n6+1) where (a,b) can be found corresponding to the 12 possibilities of kA are depicted in Figure 21.

(3.9) Definition. For i{1,,} and n we define the elements

s^i(n)ν (Si(n)) π-1(W) ( ν:Gal/W π-1(W) as in (3.1) )

It can be checked that si=s^i(0), where siπ(X,*) is as defined in (2.4).

(3.10) Corollary of lemma (3.3). (s^i(n))-1 =s^i(-n-1).

(3.11) Corollary (n=0). si-1= s^i(-1) (But in general sins^i(n) ??)

We have done the most of the work for the following result:

(3.12) Theorem. The group π-1(W) is generated by s^i(n), i{1,,}, n. A complete set of relations is:

(1) s^i(n) s^i(-n-1) for i{1,,}, n.
(2) s^i(n1) s^j(n2) s^i(n3) = s^j(nm) s^i(nm-1) s^j(nm-2) where mmij< and both sides of (2) have m factors and (n1,,nm)Nij. The sets Nijm are determined as follows: nij=njj=0 Nij=2 nij=njj=-1 Nij= { (n1,n2,n3) 3|n1 -n2+n3 {0,-1} } nij=-1,njj =-2Nij= { (n1,n2,n3,n4) 4| ( n1-n3+n4,n1-n2 +n3 ) {0,-1}2 } nij=-1,njj=-3 Nij= { (n1,,n6) 6| ( n1-n5+n6,n1 -n3+n5,n1-n2 +n3,n2-n4+n6 ) A } where A { (k1,k2,k3,k4) {0,-1}4| k1k3k4=k1 } .


We have the surjective homomorphism of semigroups ν:Gal/Wπ-1(W) (3.1). So the theorem follows from the fact that Gal/W is freely generated by the elements Si(n) and the remark that two W-orbits of galleries have the same ν-image, if and only if they are equivalent by means of the W-orbits of the cancel relations and the flip relations. And these are explicitly determined in lemma (3.3) (cancel relations) and the discussion in (3.8) (flip relations).

(3.13) Remark. In the proof of theorem (2.5) given in the next paragraph, we do not use the detailed description of the relations of the group π-1(W) given here. Instead we use properties (iii), (iv) and (v) of proposition (3.5), which we didn't use in this paragraph.

§4. Proof of the main theorem

First we outline the proof of theorem (2.5), the main theorem of this chapter. We inbed AW and Q in π(X,*) (lemma (4.1) and (4.3)). As long as the proof is not finished, it is important to distinguish between the fundamental group π(X,*) and the abstract group AW, which can be described as follows (cf. def.(2.13) and cor.(2.10)):

Add to the free product of AW and Q the push relations (2.6). The inbeddings AWπ(X,*), Qπ(X,*) induce a homomorphism AW*Qπ(X,*), which yields, after we have established the push relations (corollary (4.12)), a homomorphism of the quotient AW of AW*Q to π(X,*). The theorem will be proved, once we have shown that this homomorphism is an isomorphism.

To see this we factor the homomorphism via a quotient group G of π-1(W)*Q: AWfG hπ(X,*). We prove separately that h and f are isomorphisms (propositions (4.10) and (4.13) respectively). Here are the details:

(4.1) Lemma. The elements s1,,s as defined in (2.4) and the subgroup of π-1(W) generated by them form an Artin system. In particular AW is inbedded in π-1(W).


It follows from proposition (3.5)(ii) and (iii) that the elements siπ(X,*) satisfy the relations sisjsi= sjsisj ij, mij< and mij factors on both sides. So there is a homomorphism AWπ(X,*). The image is inside π-1(W). The fact that YY(𝕍,I,) induces a homomorphism π(Y/W) π-1(W) π(Y(𝕍,I,)/W) =AW, which is a left inverse, so the first homomorphism was injective.

(4.2) Definition. If qQ, let Tq be the curve [0,1]ttq+-1cY. Define eq{pTq}π(X,*).

Note that (see (2.4)) 𝒯i=Tαi, hence ti=eαi. In view of remark (1.2) the use of the notation eq needs justification, which is given by the following lemma:

(4.3) Lemma. The map Qqeqπ(X,*) is an injective homomorphism.


It is easy to see that the composition with π:π(X,*) induces the identity on Q.

Now we prove a prenatal form of the push relations.

(4.4) Proposition. In π(X,*) we have the following relations:

(4.5) s^i(n)· eq= eσi(q)· s^i(n+αi(q)) fori {1,,}, n,qQ.


Equivalently: (see figure 18 for this proof)

(4.6) eσi(-q)· ν(Si(n))· eq=ν (Si(n+αi(q)))

Let S be the curve in Y defined by

S(t)ρ(t)+ -1[(1-t)c+tσi(c)], whereρ: [0,1]𝕍is continuous

with the property that n<αi(ρ(12))<n+1 and ρ(0)=ρ(1)=0. Then φ ( CU(αi,n) σi(C) ) ={S}, so pS is a curve in X representing ν(Si(n)). Define curves γ1,γ2,γ3 in Y by (t[0,1]):

(4.7) γ1(t)=Tq (t)=qt+-1c γ2(t)=q+S(t) =q+ρ(t)+-1 [ (1-t)c+tσi(c) ] γ3(t)=q+σi (Tσi(-q)) =q-tq+ -1σi(c).

Then pγj represent eq, ν(Si(n)) and eσi(-q) for j=1,2 and 3 respectively. Since

γ1(1)=γ2 (0)=q+-1cand γ2(1)= γ3(0)= q+-1σi(c)

the curve γ=γ1γ2γ3 is defined and pγ represents the left hand side of (4.6). So it suffices to prove that

CU(αi,n+αi(q)) σi(C)Lγ in the sense of def. (I.1.6).

For this we check with the help of formulas (4.7) that the following hold:

(i) γ1([0,1]) γ2([0,12]) YC
(ii) γ2([12,1]) γ3([0,1]) Yσi(C)
(iii) γ2(12) Re-1 (U(αi,n+αi(q)))

Property (i) follows from Im(γ1(t))=cC and for t[0,12) we have Im(γ2(t))= (1-t)c+ σi(c) C. Moreover for t=12 the latter is contained in Mαi but then Re(γ2(12)) =q+ρ(12) M(αi,m) for all m, because we have n<αi(ρ(12))<n+1 so also:

(4.8) n+αi(q) <αi (q+ρ(12)) <n+1+ αi(q)

so αi(q+ρ(12)). This proves (i). (ii) is proved analogously and (iii) follows from (4.8).

(4.9) Definition. Let G be the group which can be described as follows: Add to the free product of π-1(W) and Q the relations (4.5).

The inclusion π-1(W)π(X,*) and the inbedding Qπ(X,*) (4.3) induce a homomorphism π-1(W)*Qπ(X,*) and by (4.4) this gives a homomorphism h:Gπ(X,*).

(4.10) Proposition. h is an isomorphism.

π-1(W)*Q π-1(W) π W G h π(X,*) π W Q

Since W=W·Q, we have π(X,*)=π-1(W)·Q so h is surjective. By the relations (4.5), which by definition (4.9) hold in G it follows that G=π-1(W)·Q. Now let xπ-1(W) and qQ and consider x·eqG. Suppose h(x·eq)=1. Then π(h(x))·π(h(eq)) is the unique decomposition of π(h(x·eq))=π(1)=1 in W, so q=π(h(eq))=0. Hence h(x)=1. Now h is already bijective on π-1(W), as can be seen form the diagram. So x=1.

Before we state the next proposition, we recall the definition of the elements si(n) for i{1,,} and n:

si(2m) timsi tim si(2m-1) timsi-1 tim

Since si,si-1AWπ-1(W) and tiQ these elements si(n) can be interpreted as elements of one of the groups AW*Q, AW, π-1(W)*Q, G and finally π(X,*). The elements s^i(n) can be seen only as elements of the last three groups (unless n=0 or -1). So the statement has only in this three cases a meaning. The statement is not true in π-1(W)*Q.

(4.11) Proposition. In G and π(X,*) we have s^i(n)=si(n)

s^i(2m)= s^i(αi(mαi))=(4.5) emαi s^i(0) emαi=(3.9) timsi tim si(2m)and s^i(2m-1)= s^i(-1+αi(mαi)) =(4.5) emαi s^i(-1) emαi=(3.11) tim si-1 tim si(2m-1).

(4.12) Corollary. In G and π(X,*) the push relations (2.6) hold.

By definition the push relations (2.6) hold in AW. The fact that they hold in G implies, that the inclusion AW*Qπ-1(W)*Q induces a homomorphism f:AWG.

(4.13) Proposition. This homomorphism f is an isomorphism.

As pointed out earlier this proposition together with proposition (4.10) proves theorem (2.5). For the proof of (4.13) we need some preparations.

(4.14) Definition. We define an action, called pseudo conjugation, of Q on π-1(W) as follows: for qQ, xπ-1(W) and π(x)wW

λq(x) eqxe-w-1(q)

Note that λq is not a homomorphism. Instead we have

(4.15) λq(xy=) λq(x) λw-1(q) (y),wπ(x)

In the same way we can define an action λ of Q on π_-1(W) as subgroup of AW, where π_:AWW is the projection induced by the projection AWWW and the identity QQW. After the establishing of the theorem π_ and π can be identified. This action of Q corresponds to an action of Q (shifting) on Gal which is defined in the following lemma:

(4.16) Lemma. An action μ of Q on Gal is induced by μq(AUB) Aq+UB. For qQ and wW we have:

(4.17) wμq= μw(q)w


Let UK(p(A,B)). We have to prove that q+UK(p(A,B)) for qQ. If A=w(C) for wW, then U=w(U(αi,n)) for some n and i{1,,}. Then q+ w(U(αi,n))= w(w-1(q)+U(αi,n)) =(by (1.7) v)=w(U(αi,n+αi(w-1(q)))) K(p(A,B)). Property (4.17) follows from w(Aq+UB)= w(A)w(q+U)w(B)= w(A)w(q)+w(U)w(B).

Since Gal/W is freely generated by Si(n) we have a homomorphism of semigroups ρ:Gal/WAW such that ρ(Si(n))=si(n). Denote the composition with GalGal/W by β, so β:GalAW.

(4.18) Lemma. the functor β is equivariant w.r.t the action λ and μ restricted to GalC, i.e. for GGal and beg(G)=C, qQ we have:

λq(β(G))= β(μq(G)).


First we prove: Suppose the lemma is true for G and G and let end(G)=w(C), wW. Then the lemma is true for G·w(G).

λq(β(G·w(G))) = λq(β(G)·β(G)) =(4.15) λq(β(G))· λw-1(q) (β(G)) = β(μq(G))· β(μw-1(q)(G)) =(4.17) β(μq(G))· β(μq(w(G))) = β(μq(G·w(G))) .

So we are done if we check the lemma for a one step gallery:

β ( μq ( CU(αi,n) σi(C) ) ) = β ( Cq+U(αi,n) σi(C) ) = β ( CU(αi,n+αi(q)) σi(C) ) = ρ(si(n+αi(q))) = si(n+αi(q))

On the other hand:

λq ( β ( CU(αi,n) σi(C) ) ) = λq(ρ(Si(n))) = λq(si(n)) = eqsi(n) e-σi(q)

and by (2.9) this equals also si(n+αi(q)).

(4.19) Proposition. There exists a unique homomorphism g:π-1(W)AW such that we have ρ=gν. Note that g(s^i(n))=si(n).

Gal/W νρ π-1(W) g AW


It is enough to check that the elements si(n)AW satisfy the cancel relations and the flip relations. First the cancel relations:

si(2m) si(-2-1)= timsi tim ti-m si-1 ti-m=1and si(2m-1) si(-2m+1-1) =timsi-1 timti-m siti-m=1

Now the flip relations: Suppose i,j{1,,} and mmij< and (n1,,nm)Nij. Let xsi(n1)si(n2)si(n3) and ysj(nm)si(nm-1) (both m factors). We have to prove: x=y. Let qQ be as exists by (3.5)(v), i.e. 0q+U. Then it follows of lemma (4.18) and proposition (3.15)(iv) that λq(x), λq(y) ? AW. Since f(x)=f(y), because the flip relations hold by definition in G, we have λq(f(x))=λq(f(y)), hence also f(λq(x))=f(λq(y)). But f is injective on AW as follows from the diagram, so λq(x)=λq(y). Hence x=y.

AW AW*Q AW f G h π(X,*)

Proof of proposition (4.13).

The homomorphism g:π-1(W)AW induces a homomorphism π-1(W)*QAW and this induces a homomorphism GAW which is an inverse of f:AWG as easily can be checked on various generators.

Now theorem (2.5) is completely proved.

§5. Affine Artin groups

In this paragraph we make first some remarks about affine or parabolic linear Coxeter groups. This makes clear why the result of Nguyen Viet Dung [Ngu1983] is indeed a special case of theorem (II.3.8). Next we establish an isomorphism between an extended Artin group of finite type and a certain affine Artin group.

(5.1) Definition. A Coxeter group is called affine or parabolic, if it is isomorphic to a group Wa of affine transformations of an affine space E, such that Wa acts properly discontinuously on E and is generated by orthogonal reflections.

It can be proved (Bourbaki [Bou1968]) that such a group is generated by the reflections in the walls of a chamber with respect to the reflection hyperplanes of the reflections in the group. The group forms with these generators a Coxeter system.

If one want to construct a space of regular orbits in the situation of definition (5.1) there are various possibilities (depending on a parameter ξ*), which we will encounter in (5.3). One possibility (corresponding to ξ=1+-1) is obvious by the affine nature of E: Let EE×E and let the group Wa act on E by acting on both components. The complexified reflection hyperplanes are then of the form M+-1M, where M is a real affine reflection hyperplane of Wa. This is the approach of Nguyen Viet Dung [Ngu1983].

It turns out that it is not important to distinguish between these various complexifications, because they lead to homeomorphic regular orbit spaces, as we will see in (5.3).

(5.2) Definition. Let Wa be a linear Coxeter group acting on a real vector space 𝕍. Wa is called an affine (or parabolic) linear Coxeter group is there is an invariant hyperplane 𝕍𝕍, with an invariant innerproduct and 𝕍I={0}.

Let ψ(𝕍)* such that 𝕍=ψ-1(0) and Iψ-1((0,)). Then Eψ-1(1) is also invariant and Wa acts on E as an affine Coxeter group (in particular I=ψ-1((0,))) as described in def. (5.1). Conversely given an affine Coxeter group as in (5.1) we can form 𝕍E×, considering E as a vector space. Now each (affine) reflection in E extends uniquely to a linear reflection in 𝕍 and in this way we get a parabolic linear Coxeter group. The form ψ can be defined by ψ(x,t)t.

(5.3) Complexification. Let Wa be an affine linear Coxeter group in a space 𝕍 and let ψ(𝕍)* be as before. Let Y and X be the space of regular points and the regular orbit space respectively of Wa as defined in (II.3.2) and (II.3.3).

Let ξ* and consider Eξψ-1(ξ). Let YξEξ- MM and XξYξ/Wa. The map Y1zξzYξ is a homeomorphism equivariant under the action of Wa, so Xξ is homeomorphic to X1. On the other hand for Im(ξ)>0, we have that Yξ is a deformation retract of Y, because if zY, then zξξψ(z)·zYξ and [z,zξ]Y. This deformation retraction is also equivariant for the action of Wa, so Xξ is a deformation retract of X.

(5.4) In particular this shows that the result of Nguyen Viet Dung, which in fact says that the fundamental group of X1+-1 is the Artin group associated to Wa, is a special case of theorem (II.3.8).

We apply these ideas to the situation of an extended Coxeter group W of finite type, i.e. W is finite. Then the number of roots is finite and we have a classical root system. We assume this to be irreducible. Let α0 be the longest dual root, α0 the corresponding root. The corresponding reflection σα0W is defined by for v𝕍: σα0(v) v-α0(v)α0. Define also the affine reflection σ0W by σ0eα0σα0. Let Wa be the group of affine transformations of 𝕍 generated by {σ0,σ1,,σ}. Then it is known (Bourbakl [Bou1968] Ch.VI §2) that (Wa,{σ0,σ1,,σ}) is a (parabolic) Coxeter system, C{x𝕍|α0(x)1} is a fundamental domain of Wa and Wa=W. We want to establish an isomorphism between the (parabolic) Artin group AWa associated to Wa and the extended Artin group AW:

(5.5) Theorem. Let B={α1,,α}, :BB be the root basis of a finite irreducible root system. Let α0 be the longest dual root, α0 the corresponding root, σα0 the corresponding reflection in the Weyl group W.

Let (AWa,{s0,s1,,s}) be the Artin system associated to the Coxeter system (Wa,{σ0,σ1,,σ}), where σ0eα0σα0W.

Then AWa and AW are isomorphic and an isomorphism is induced by:

(5.6) si si fori=1,, s0 t0sα0-1

where t0eα0 and sα0 is the image of σα0 under the injection WAW discussed in (II.3.?).


From the fact that W is finite it follows that W is an affine Coxeter group acting on 𝕍 as discussed in the beginning of this paragraph. We construct the (affine) linear Coxeter system (Wa,{σ0,σ1,,σ}) as suggested there. Put

𝕍 𝕍× vi (αi,0), ϕi(v,t) αi(v) i=1,, v0 (-α0,0), ϕ0(v,t) t-α0(v)

And as usual: σi(v,t) (v,t)- ϕi(v,t)vi for i=1,,. Now Wa leaves invariant Eξ𝕍C×{ξ} 𝕍×=(𝕍) for each ξ*. Let us in particular study this action for ξ=1.

For i=1,, we have:

σi(v,1) = (v,1)- αi(v,1) vi = (v,1)-αi (v)(αi,0) = (σi(v),1) σ0(v,1) = (v,1)-α0 (v,1)v0 = (v,1)- (1-α0(v)) (-α0,0) = ( v-α0(v)α0 +α0,1 ) = ( σα0(v)+ α0,1 ) = ( σ0(v),1 )

So the action of Wa on 𝕍×{1}𝕍 corresponds with the action of W on 𝕍. Hence XX1. So by the reasoning in (5.3) we have XX, hence AWaAW.

We check that an isomorphism is explicitly given by (5.6). Consider the diagram

YgY p p Y-1 ($$) Y11 X f X

where ($) is the deformation retraction defined in (5.3) and ($$) is the multiplication with --1. The basepoint of X is p(-1(c,1)), its image under f:f p(-1(c,1)) =p(g(-1(c,1))) =p(c). But the basepoint of X is p(-1c). So we define λ1[-1c,c] and then pλ1 is a curve from the basepoint of X to the f-image of the basepoint of X. So an explicit isomorphism π(X,p(-1(c,1))) π(X,p(-1c)) is given by

(5.7) {pγ} {pλ1} {pgγ} {pλ1}-1

Now let γ be a curve in Y such that pγ represents the element (s0)-1, so for instance we can take γ(t) ρ(t)v0+ -1 { (1-t)(c,1)+ tσ0(c,1) } with ρ(12)<0. Let γgγ. Then γ(t)= -1ρ(t)α0+ (1-t)c+ tσ0(c) which is a curve from c to σ0(c)=σα0(c)+α0. Now we have to study the curve μ=λ1γλ2-1, where λ2=[-1σα0(c)+α0σα0(c)+α0]. For note that pλ1=pλ2, so pμ represents the right hand side of (5.7). The curve μ is illustrated in Figure 22, where the root system B2 is shown.

α0 α1 α2 σ0(c) c c C σα0(c) Real part ofλ1γλ2-1 Imaginary part ofλ1γλ2-1 Figure 22.

Now we make four remarks about this curve μ. These remarks are sufficient to describe its homotopy type:

(1) Im(μ) is a curve from c to σα0(c).
(2) Re(μ) is a curve from 0 to α0 and except for the begin and end point there is only one other value of the parameter, say t=12 such that the real part is on some hyperplane NpM, M.
(3) We have Im(μ(12))=ρ(12)α0. Since ρ(12)<0 this means that the imaginary part has already passed by all the hyperplanes separating C and σα0(C).
(4) While Im(μ) is passing by the hyperplanes separating C and σα0(C) the real part is in C{x𝕍|α0(x)<1}.

From these considerations it follows that pμ is a loop in X representing the element sα0e-α0AW. So the image of s0 is eα0sα0-1= t0sα0-1. A similar reasoning gives that the image of siAWa is siAW.

(5.8) Corollary. If AW is an extended Artin group of rank , such that W is finite and irreducible, then AW has a finite presentation with +1 generators.

The conclusion of this corollary is true under some other conditions too:

(5.9) Theorem. If AW is an extended Artin group of rank with nij{0,-1} for all i,j{1,,} and with a connected Dynkin diagram, then AW has a finite presentation with +1 generators.


If nij=-1 then sitjsi=titj (2.9), so then ti can be eliminated. Since the Dynkin diagram is connected all but one of the generators t1,,t can be eliminated.

But there is an important difference between (5.8) and (5.9). The relations of the presentation meant in (5.8) are pretty simple, because they are the relations of an Artin group. But the relations become very complicated if one follows the procedure of the proof of (5.9).

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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