Chapter III. Extended Artin groups

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

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 ${}^{\vee }:B\to {𝕍}^{*},$ $\alpha \to {\alpha }^{\vee },$ $\alpha \in B$ is called a root basis if the following conditions are satisfied:

(i)	$B$ (respectively $B^{\vee }\text{)}$ is a linearly independent subset of $𝕍$ (resp. ${𝕍}^{*}\text{).}$
(ii)	${\alpha }^{\vee }\left(\alpha \right)=2$ for all $\alpha \in B\text{.}$
(iii)	${\beta }^{\vee }\left(\alpha \right)$ is a nonpositive integer for $\alpha ,\beta \in B,$ $\alpha \ne \beta \text{.}$
(iv)	${\beta }^{\vee }\left(\alpha \right)=0$ implies ${\alpha }^{\vee }\left(\beta \right)=0\text{.}$

We order the elements of $B=\{{\alpha }_1,\dots ,{\alpha }_{\ell }\},$ $\ell =\left|B\right|\text{.}$ The Cartan matrix $N$ has entries defined by $n_{ij}≔{\alpha }_i^{\vee }\left({\alpha }_j\right)$ (This deviates from the convention in [Lek1981] and [Loo1980], where $n_{ij}={\alpha }_j^{\vee }\left({\alpha }_i\right)).$

Two root bases $({𝕍}_1,B_1)$ and $({𝕍}_2,B_2)$ are said to be isomorphic if they are so in the sense of (II.2), i.e. if there exists a linear isomorphism $f:{𝕍}_1\to {𝕍}_2$ such that $f\left(B_1\right)=f\left(B_2\right)$ and $f^{*}\left(B_2^{\vee }\right)=f^{*}\left(B_1^{\vee }\right)\text{.}$

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 $\text{dim}\left(𝕍\right)\ge \ell +\text{corank}\left(N\right)$ 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 $(C,\{{\sigma }_1,\dots ,{\sigma }_{\ell }\})$ is a Vinberg pair, where

$$\begin{array}{cc}\text{(1.1)}& \begin{array}{ccc}{\sigma }_i\left(x\right)& ≔& x-{\alpha }_i^{\vee }\left(x\right){\alpha }_i\text{and}\\ C& ≔& \{x\in 𝕍\hspace{0.17em}|\hspace{0.17em}{\alpha }_i^{\vee }\left(x\right)>0\}\end{array}\end{array}$$

So if as before $W$ denotes the subgroup of $\text{Gl}\left(𝕍\right)$ generated by the reflections $\{{\sigma }_1,\dots ,{\sigma }_{\ell }\}$ and $I≔\bigcup _{w\in W}w\left(\bar{C}\right)$ 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 $Q≔{⨁}_{i=1}^{\ell }\mathbb{Z}{\alpha }_i,$ the so-called (root) lattice. Since ${\sigma }_i\left(Q\right)\subset Q,$ $W$ acts on $Q\text{.}$ Let $\stackrel{\sim }{W}$ be the group of affine transformations of $𝕍$ generated by $W$ and $Q\text{.}$ 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 $q\in Q$ as $e^q\text{.}$ For instance $Q\subset \stackrel{\sim }{W}\text{.}$ In this case we denote the elments $e^{{\alpha }_i}$ by ${\tau }_i\text{.}$

Since for $w\in W,q\in Q$ and $x\in 𝕍$ we have:

$$w·e^q\left(x\right)=w(q+x)=w\left(q\right)+w\left(x\right)=e^{w\left(q\right)}·w\left(x\right)$$

we see that $\stackrel{\sim }{W}$ is the semidirect product of $W$ and $Q\text{.}$ 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 $\stackrel{\sim }{W}$ can be described as follows:
Add to the free product of $W$ and $Q$ the following relations:

$${\sigma }_i·e^q=e^{{\sigma }_i\left(q\right)}·{\sigma }_{i}i\in \{1,\dots ,\ell \},\hspace{0.17em}q\in Q\text{.}$$

(1.4) Proposition. The group $\stackrel{\sim }{W}$ is generated by ${\sigma }_1,\dots ,{\sigma }_{\ell },$ ${\tau }_1,\dots ,{\tau }_{\ell },$ where ${\tau }_i≔e^{{\alpha }_i}\text{.}$ A complete set of relations is $\text{(}i,j\in \{1,\dots ,\ell \}\text{):}$

1. ${\left({\sigma }_i{\sigma }_j\right)}^{m_{ij}}=1$ for $m_{ij}<\infty \text{.}$
2. ${\tau }_i{\tau }_j={\tau }_j{\tau }_i\text{.}$
3. ${\sigma }_i{\tau }_j={\tau }_j{{\tau }_i}^{-n_{ij}}{\sigma }_i\text{.}$

(1.5) Proposition. If $\left|W\right|<\infty ,$ then $\stackrel{\sim }{W}$ is an affine Coxeter group (5.1). Then it acts simply transitively on the chambers w.r.t. the reflection hyperplanes (mirrors) of $\stackrel{\sim }{W}\text{.}$

		Proof.
	See Bourbaki [Bou1968] Ch. V §3. $\square$

The group $W$ acts on ${𝕍}^{*}$ by the rule $w\left(\tau \right)\left(x\right)≔\tau \left(w^{-1}\left(x\right)\right),$ for $x\in 𝕍,$ $w\in W$ and $\tau \in {𝕍}^{*}\text{.}$ Let $R≔W·B$ (respectively $R^{\vee }≔W·B^{\vee }\text{)}$ denote the set of roots (respectively dual roots). This construction is called a generalised root system.

(1.6) Definition. For ${\alpha }^{\vee }\in R^{\vee }$ and $n\in \mathbb{Z}$ we introduce:

$$\begin{array}{ccc}M({\alpha }^{\vee },n)& ≔& \{x\in 𝕍\hspace{0.17em}|\hspace{0.17em}{\alpha }^{\vee }\left(x\right)=n\},\\ M_{{\alpha }^{\vee }}& ≔& M({\alpha }^{\vee },0),\\ U({\alpha }^{\vee },n)& ≔& \{x\in 𝕍\hspace{0.17em}|\hspace{0.17em}n<{\alpha }^{\vee }\left(x\right)<n+1\},\\ U({\alpha }^{\vee },+)& ≔& \{x\in 𝕍\hspace{0.17em}|\hspace{0.17em}{\alpha }^{\vee }\left(x\right)>0\},\\ ℳ& ≔& \left\{M_{{\alpha }^{\vee }}\hspace{0.17em}\right|\hspace{0.17em}{\alpha }^{\vee }\in R^{\vee }\},\text{the set of mirrors of}\hspace{0.17em}W,\\ p_{{\alpha }^{\vee }}& ≔& \left\{M({\alpha }^{\vee },n)\hspace{0.17em}\right|\hspace{0.17em}n\in \mathbb{Z}\},\\ p& :& ℳ\ni M_{{\alpha }^{\vee }}\to p_{{\alpha }^{\vee }}\text{.}\end{array}$$

(1.7) Proposition. Let ${\alpha }^{\vee }\in R^{\vee },$ $w\in W$ and $n\in \mathbb{Z}\text{.}$ Then

i)	The quadruple $(𝕍,\stackrel{\circ }{I},ℳ,p)$ is a hyperplane system (I.2.7),
ii)	$w\left(M({\alpha }^{\vee },n)\right)=M(w\left({\alpha }^{\vee }\right),n),$
iii)	$w\left(U({\alpha }^{\vee },n)\right)=U(w\left({\alpha }^{\vee }\right),n),$
iv)	$w\left(U({\alpha }^{\vee },+)\right)=U(w\left({\alpha }^{\vee }\right),+),$
v)	$q+M({\alpha }^{\vee },n)=M({\alpha }^{\vee },n+{\alpha }^{\vee }\left(q\right)),$
vi)	$q+U({\alpha }^{\vee },n)=U({\alpha }^{\vee },n+{\alpha }^{\vee }\left(q\right))\text{.}$

		Proof.
	Straightforward. For instance vi): $x\in q+U({\alpha }^{\vee },n)\iff x-q\in U({\alpha }^{\vee },n)\iff n<{\alpha }^{\vee }(x-q)<n+1\iff n+{\alpha }^{\vee }\left(q\right)<{\alpha }^{\vee }\left(x\right)<n+{\alpha }^{\vee }\left(q\right)+1\text{.}$ $\square$

§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 $\stackrel{\sim }{W}\text{.}$ As in (II.3) the group $W$ acts properly discontinuously on the domain $\Omega ≔{\text{Im}}^{-1}\left(\stackrel{\circ }{I}\right)\text{.}$ Also $\stackrel{\sim }{W}$ acts properly discontinuously on $\Omega ,$ where the action of $Q$ is given by $x+\sqrt{-1}y\to x+q+\sqrt{-1}y$ for $q\in Q,$ i.e. $Q$ affects the real plane only.

In $\Omega$ the points in regular orbits or the regular points, ??? are by definition the points with trivial stabiliser (of the $\stackrel{\sim }{W}\text{-action}$ ???) are the points of the space

$$\begin{array}{cc}\text{(2.1)}& Y≔\Omega -\left\{\text{reflection hyperplanes of}\hspace{0.17em}\stackrel{\sim }{W}\right\}\text{.}\end{array}$$

This follows from:

(2.2) Lemma. The ??? of non regular points in $\Omega$ is the union of the reflection hyperplanes of $\stackrel{\sim }{W}\text{.}$ The reflection hyperplanes are the sets of the form ??? for some ${\alpha }^{\vee }\in R^{\vee }$ and $n\in \mathbb{Z}\text{.}$

		Proof.
	See Looijenga [Loo1980] lemma 2.17. $\square$

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

$$\begin{array}{cc}\text{(2.3)}& X≔Y/\stackrel{\sim }{W}\text{and let}p:Y\to X\text{denote the projection.}\end{array}$$

In the fundamental chamber $C$ (1.1) we choose a point $c\text{.}$ Now $\sqrt{-1}c\in Y$ (from (2.2)) and we take $*≔p\left(\sqrt{-1}c\right)$ as a base point for $X\text{.}$

(2.4) Now we define elements $s_1,\dots ,s_{\ell },t_1,\dots ,t_{\ell }\in \pi (X,*)\text{.}$ For $i=1,\dots ,\ell$ define:

$$\begin{array}{cccc}{𝒮}_i:& [0,1]& ⟶& Y\text{by}\\ & t& ⟼& \delta \left(t\right){\alpha }_i+\sqrt{-1}((1-t)c+t{\sigma }_i\left(c\right))\end{array}$$

where $\delta :[0,1]\to [0,\frac{1}{2})$ is continuous and $\delta \left(\frac{1}{2}\right)\ne 0,$ e.g. $\delta \left(t\right)=\frac{1}{4}\text{sin}\left(\pi t\right),$ see figure 16.

$$\begin{array}{cccc}{𝒯}_i:& [0,1]& ⟶& Y\text{by}\\ & t& ⟼& t{\alpha }_i+\sqrt{-1}c\text{.}\end{array}$$

Now $p\circ {𝒮}_i$ and $p\circ {𝒯}_i$ are closed curves (loops) in $X$ and they represent by definition the classes $s_i$ and $t_i$ respectively.

$$\begin{array}{c} 12 0 1 12 \\ \text{Figure 16.}\end{array}$$

Now we can state the main theorem of this chapter:

(2.5) Theorem. The fundamental group $\pi (X,*)$ of the regular orbit space $X$ of an extended Coxeter group $\stackrel{\sim }{W}$ is generated by two subgroups isomorphic to $A_W$ and $Q$ respectively. In turn these groups are generated by $\{s_1,\dots ,s_{\ell }\}$ and $\{t_1,\dots ,t_{\ell }\}$ respectively, where $s_i$ and $t_i$ are as above.

Furthermore the following relations hold:

$$\begin{array}{cc}\text{(2.6)}& {s_i}^{\left(n\right)}<mtext>.</mtext>e^q=e^{{\sigma }_i\left(q\right)}·{s_i}^{(n+{\alpha }_i^{\vee }\left(q\right))}\end{array}$$

for $i=1,\dots ,\ell ,$ $q\in Q$ and $n\in \mathbb{Z},$ where

$$\begin{array}{cc}\text{(2.7)}& s_i^{\left(2m\right)}≔t_i^m{s}_i{t}_i^m\text{and}{s_i}^{(2m-1)}≔t_i^m{s}_i^{-1}{t}_i^m\text{for}m\in \mathbb{Z}\text{.}\end{array}$$

This is a complete set of relations.

(2.8) Remark. The relations (2.6) are called push relations, because they tell how to "push" $e^q$ through $s_i^{\left(n\right)}\text{.}$ This "pushing" seems to be from the right to the left, however we also have:

$$\begin{array}{cc}\text{(2.9)}& e^q·s_i^{\left(n\right)}=s_i^{(n+{\alpha }_i^{\vee }\left(q\right))}·e^{{\sigma }_i\left(q\right)}\text{.}\end{array}$$

		Proof of (2.9).
	By (2.6) the right hand side of (2.9) equals $e^{{\sigma }_i\left({\sigma }_i\left(q\right)\right)}·{s_i}^{(n+{\alpha }_i^{\vee }\left(q\right)+{\alpha }_i^{\vee }\left({\sigma }_i\left(q\right)\right))}$ and this equals $e^q·s_i^{\left(n\right)},$ for ${\alpha }_i^{\vee }(q+{\sigma }_i\left(q\right))={\alpha }_i^{\vee }(2q-{\alpha }_i^{\vee }\left(q\right){\alpha }_i)=2{\alpha }_i^{\vee }\left(q\right)-{\alpha }_i^{\vee }\left(q\right){\alpha }_i^{\vee }\left({\alpha }_i\right)=0\text{.}$ $\square$

We will prove theorem (2.5) in §3 and §4. Now we first illustrate the elements $s_i^{\left(n\right)}$ and the push relations:

$$\begin{array}{cc} 12U(αi∨,n) αi & Mαi∨ C c \\ \text{real part of}\hspace{0.17em}{s}_i^{\left(n\right)}& \text{imaginary part of}\hspace{0.17em}{s}_i^{\left(n\right)}\\ \multicolumn{2}{c}{\text{Figure 17.}}\end{array}$$

Figure 17 depicts a curve in $Y,$ which is the lift of a typical representative of $s_i^{\left(n\right)}\text{.}$ The imaginary part moves from $c$ to ${\sigma }_i\left(c\right)\text{.}$ The real part starts and ends in the origin, but when the imaginary part is on $M_{{\alpha }_i^{\vee }},$ the real part has to be in the layer $U({\alpha }_i^{\vee },n)≔\{x\in V\hspace{0.17em}|\hspace{0.17em}n<a_i^{\vee }\left(x\right)<n+1\}\text{.}$

$$\begin{array}{c} q+U(αi∨,n)=U(αi∨,n+αi∨(q)) } } } n αi∨(q) n+αi∨(q) 0 q \\ \text{Figure 18.}\end{array}$$

Figure 18 depicts the push relations $s_i^{\left(n\right)}·e^q=e^{{\sigma }_i\left(q\right)}·s_i^{(n+{\alpha }_i^{\vee }\left(q\right))}\text{.}$ A lift of a typical representative of $s_i^{\left(n\right)}·e^q$ can be described as follows: First the real part moves from the origin to $q\text{.}$ Then the curve described above takes place, the real part being shifted over $q\text{.}$ So now, if the imaginary part is on $M_{{\alpha }_i^{\vee }},$ the real part is in the layer $q+U({\alpha }_i^{\vee },n)=U({\alpha }_i^{\vee },n+{\alpha }_i^{\vee }\left(q\right))\text{.}$ Therefore intuitively we can see that we can also start $s_i^{(n+{\alpha }_i^{\vee }\left(q\right))}\text{.}$ The element $e^{{\sigma }_i\left(q\right)}$ is then necessary to take care, that the real part ends in the correct point, namely $q\text{.}$

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

As a second reformulation of the main result (2.5) we give an explicit presentation of $\pi (X,*)\text{.}$

(2.11) Corollary (cf. prop. (1.4)). The group $\pi (X,*)$ is generated by the elements $s_1,\dots ,s_{\ell },t_1,\dots ,t_{\ell },$ a complete set of relations is:

1b.	$s_i{s}_j{s}_i\dots =s_j{s}_i{s}_j\dots i\ne j,m_{ij}<\infty ,\hspace{0.17em}{m}_{ij}$ factors both sides.
2.	$t_i{t}_j=t_j{t}_i$
3b.	$s_i{t}_j=t_j{t}_i^r{s}_i{t}_i^{-r}{n}_{ij}$ even and $n_{ij}=-2r$
3c.	$s_i{t}_j=t_j{t}_i^{r+1}{s}_i^{-1}{t}_i^{-r}{n}_{ij}$ odd and $n_{ij}=-2r-1$

where $i,j\in \{1,\dots ,\ell \}\text{.}$

In fact these corollaries are equivalent to the theorem.

		Proof of (2.5) $\iff$ (2.11).
	Relations 3b. and 3c. are special cases of the push relations (2.6). Namely, if $n=0$ and $q={\alpha }_j$ then: $$s_i{t}_j=t_i^{\left(0\right)}·e^{{\alpha }_j}=e^{{\sigma }_i\left({\alpha }_j\right)}·{s_i}^{\left({\alpha }_i^{\vee }\left({\alpha }_j\right)\right)}$$ Now ${\sigma }_i\left({\alpha }_j\right)={\alpha }_j-{\alpha }_i^{\vee }\left({\alpha }_j\right){\alpha }_i={\alpha }_j-n_{ij}{\alpha }_i,$ so $e^{{\sigma }_i\left({\alpha }_j\right)}=t_j{t}_i^{-n_{ij}}\text{.}$ If $n_{ij}≔{\alpha }_i^{\vee }\left({\alpha }_j\right)$ is even, say $-n_{ij}=2r,$ then $$s_i{t}_j=t_j{t}_i^{2r}{t}_i^{-r}{s}_i{t}_i^{-r}=t_j{t}_i^r{s}_i{t}_i^{-r}\text{.}$$ And if $n_{ij}$ is odd, say $-n_{ij}=2r+1,$ then $$s_i{t}_j=t_j{t}_i^{2r+1}{t}_i^{-r}{s}_i^{-1}{t}_i^{-r}=t_j{t}_i^{r+1}{s}_i^{-1}{t}_i^{-r}\text{.}$$ Conversely it is easy to see that the push relations (2.6) follow from these special cases and the commutation relations 2. $\square$

Relations 3b. and 3c. say how one can "push $t_j$ through $s_i\text{".}$ But one cannot push $t_i$ through $s_i,$ because for $i=j$ we have $-n_{ij}=-2,$ so $r=-1$ and 3b. becomes $s_i{t}_i=s_i{t}_i\text{.}$ This in contrast to $\stackrel{\sim }{W}$ where the relation 3a. ${\sigma }_i{\tau }_j={\tau }_i^{-1}{\sigma }_j$ is valid.

In the special case that $n_{ij}\in \{0,-1\}$ for all $i\ne j,$ we get a complete set of relations:

$$\begin{array}{cc}\text{(2.12)}& \begin{array}{cccc}{s}_i{s}_j& =& s_j{s}_i& \}\text{for}\hspace{0.17em}{n}_{ij}=0\text{.}\\ s_i{t}_j& =& t_j{s}_i\\ t_i{t}_j& =& t_j{t}_i& \text{for all}\hspace{0.17em}i,j\in \{1,\dots ,\ell \}\text{.}\\ s_i{s}_j{s}_i& =& s_j{s}_i{s}_j& \}\text{for}\hspace{0.17em}{n}_{ij}=-1\text{.}\\ s_i{t}_j{s}_i& =& t_i{t}_j\end{array}\end{array}$$

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 $\stackrel{\sim }{W},$ changing it in a suitable equivalent presentation and then deleting some relations. In fact this describes precisely the homomorphism $\pi :\pi (X,*)\to \stackrel{\sim }{W}\text{.}$

A presentation of $\stackrel{\sim }{W}$ (1.4):	Another presentation of $\stackrel{\sim }{W}:$
generators: ${\sigma }_1,\dots ,{\sigma }_{\ell },{\tau }_1,\dots ,{\tau }_{\ell }$	generators: ${\sigma }_1,\dots ,{\sigma }_{\ell },{\tau }_1,\dots ,{\tau }_{\ell }$
relations:	relations:
1. ${\left({\sigma }_i{\sigma }_j\right)}^{m_{ij}}=1$	1a. ${\sigma }_i^2=1$ 1b. ${\sigma }_i{\sigma }_j{\sigma }_i\dots ={\sigma }_j{\sigma }_i{\sigma }_j\dots ,$ $i\ne j,$ $m_{ij}<\infty$ $m_{ij}$ factors
2. ${\tau }_i{\tau }_j={\tau }_j{\tau }_i$	2. ${\tau }_i{\tau }_j={\tau }_j{\tau }_i$
3. ${\sigma }_i{\tau }_j={\tau }_j{\tau }_i^{-n_{ij}}{\sigma }_i$	3a. $(i=j)$ ${\sigma }_i{\tau }_i={\tau }_i^{-1}{\sigma }_i$ 3b. $(-n_{ij}=2r)$ ${\sigma }_i{\tau }_j={\tau }_j{\tau }_j^r{\sigma }_i{\tau }_i^{-r}$ 3c. $(-n_{ij}=2r+1)$ ${\sigma }_i{\tau }_j={\tau }_j{\tau }_j^{r+1}{\sigma }_i^{-1}{\sigma }_i{\tau }_i^{-r}$

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

1a.	${\sigma }_i^2=1$	$\text{(}W$ changes into $A_W\text{)}$
3a.	${\sigma }_i{\tau }_i={\tau }_i^{-1}{\sigma }_j$	(pushing $\tau i$ through ${\sigma }_i$ is no longer allowed)

(2.13) Definition. We call the abstract group defined by the presentation (2.11) the extended Artin group $A_{\stackrel{\sim }{W}},$ associated to the extended Coxeter group $\stackrel{\sim }{W}\text{.}$ The number $\ell$ 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 ${\pi }^{-1}\left(W\right)$

Let $\pi :\pi (X,*)\to \stackrel{\sim }{W}$ be the canonical surjective homomorphism. In this paragraph we derive a presentation of the subgroup it ${\pi }^{-1}\left(W\right)$ of $\pi (X,*),$ as a first step in the proof of the main theorem (2.5) and because it is interesting in itself.

The space $Y$ equals $Y(𝕍,\stackrel{\circ }{I},ℳ,p),$ where $ℳ$ and $p$ are as defined in (1.6). Recall that below (2.3) we have fixed a point $c\in C\text{.}$ For each $w\in W$ we define a $a_{w\left(C\right)}≔w\left(c\right)\in w\left(C\right),$ 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 $\phi :\text{Gal}\to \Pi$ gives us a surjective homomorphism of semigroups $\phi /W:\text{Gal}/W\to \Pi /W\text{.}$

The cover $Y/W\to X$ induces an injection $\pi (Y/W)\to \pi \left(X\right)$ of fundamental groups and the image is exactly ${\pi }^{-1}\left(W\right)\text{.}$ Hence ${\pi }^{-1}\left(W\right)$ is isomorphic to $\pi (Y/W)\text{.}$ Now $\pi (Y/W)$ is canonically isomorphic to $\Pi /W,$ as can be seen in the same way as (II.3.16). Let $\nu$ be the surjective homomorphism obtained by $\phi /W$ and this last isomorphism:

$$\begin{array}{cc}\text{(3.1)}& \begin{array}{ccccccc}\text{Gal}/W& \multicolumn{3}{c}{\stackrel{\nu }{⟶}}& {\pi }^{-1}\left(W\right)& \subset & \pi (X,*)\\ \multicolumn{2}{c}{\phi /W\nwarrow }& & \multicolumn{2}{c}{\nearrow \cong }\\ & & \Pi /W\end{array}\end{array}$$

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

$$\begin{array}{cc}\text{(3.2)}& C^{U({\alpha }_i^{\vee },n)}{\sigma }_i\left(C\right)\text{where}\hspace{0.17em}i\in \{1,\dots ,\ell \}\hspace{0.17em}\text{and}\hspace{0.17em}n\in \mathbb{Z}\text{.}\end{array}$$

We denote the $W\text{-orbit}$ of (3.2) by $S_i^{\left(n\right)}\text{.}$

Now we study the $W\text{-orbits}$ of the cancel and flip relations.

(3.3) Lemma. The $W\text{-orbits}$ of the cancel relations are:

$$(S_i^{\left(n\right)}{S}_i^{(-n-1)},\varnothing )\text{for}\hspace{0.17em}i=1,\dots ,\ell \hspace{0.17em}\text{and}\hspace{0.17em}n\in \mathbb{Z}\text{.}$$

		Proof.
	A typical cancel relation is of the form: $$\begin{array}{cc}\text{(\$)}& (C^{U({\alpha }_i^{\vee },n)}{{\sigma }_i\left(C\right)}^{U({\alpha }_i^{\vee },n)}C,{\varnothing }_C)\end{array}$$ Since ${\sigma }_i\left(U({\alpha }_i,n)\right)=U(-{\alpha }_i,n)=\{x\in V\hspace{0.17em}|\hspace{0.17em}n<-{\alpha }_i^{\vee }\left(x\right)<n+1\}=\{x\in 𝕍\hspace{0.17em}|\hspace{0.17em}-n-1<{\alpha }_i^{\vee }\left(x\right)<-n\}=U({\alpha }_i,-n-1),$ the $W\text{-orbit}$ of the left element of ($) is $S_i^{\left(n\right)}{S}_i^{(-n-1)}\text{.}$ $\square$

(3.4) Definition. If $(G,G\prime )\subset \text{Gal}\times \text{Gal}$ is a flip relation, then we call $G$ flippable and $G\prime$ its flipped form. We use the same terminology for $W\text{-orbits.}$

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

(3.5) Proposition.

(i)	A gallery $G=C^{U_1}{C_1}^{U_2}\dots {}^{U_m}{C}_m$ is flippable if and only if there are $i,j\in \{1,\dots ,\ell \}$ with $m=m_{ij}$ (so $m_{ij}<\infty \text{)}$ $i\ne j$ and the $W\text{-orbit}$ of $G$ is of the form:

$$\begin{array}{cc}\text{(3.6)}& g=S_i^{\left(n_1\right)}{S}_j^{\left(n_2\right)}{S}_i^{\left(n_3\right)}\dots \text{(}m\hspace{0.17em}\text{factors)}\text{for some}{n}_1,\dots ,n_m\in \mathbb{Z}\text{and}U≔U_1\cap \dots \cap U_m\ne \varnothing \end{array}$$

(ii)	In this case the flipped form of the $W\text{-orbit}$ $g$ is:

$$\begin{array}{cc}\text{(3.7)}& S_j^{\left(n_m\right)}{S}_i^{\left(n_{m-1}\right)}{S}_j^{\left(n_{m-2}\right)}\dots \text{(}m\hspace{0.17em}\text{factors)}\end{array}$$

(iii)	If $n_1=n_2=\dots =n_m=0$ and $g$ of the form (3.6), then $0\in \bar{U}\text{.}$ In particular $U\ne \varnothing ,$ hence $G$ and $g$ are flippable.
(iv)	If the set $U$ has the property that $0\in \bar{U},$ then $n_r\in \{0,-1\}$ for $r=1,\dots ,m\text{.}$
(v)	If $G$ is flippable, then there is a $q\in Q$ such that $0\in q+\bar{U}\text{.}$

		Proof.
	(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 $U\ne \varnothing$ is part of the definition of flippable. The converse is obvious. (ii) From the fact that $g$ is the $W\text{-orbit}$ of $G$ we know that $$\begin{array}{cc}\text{(£)}& U_r=U({\alpha }^{\vee }\left(r\right),n_r)r=1,\dots ,m\text{where}\hspace{0.17em}{\alpha }^{\vee }\left(r\right)\hspace{0.17em}\text{is some dual root of}\hspace{0.17em}{R}_{\{i,j\}}^{\vee }\text{.}\end{array}$$ (In fact we have ${\alpha }^{\vee }\left(1\right)={\alpha }_i^{\vee },$ ${\alpha }^{\vee }\left(2\right)={\sigma }_i\left({\alpha }_j^{\vee }\right),$ ${\alpha }^{\vee }\left(3\right)={\sigma }_i{\sigma }_j\left({\alpha }_i^{\vee }\right),$ etc.) Suppose the flipped form of $g$ is $S_j^{\left(n_m^{\prime }\right)}{S}_i^{\left(n_{m-1}^{\prime }\right)}{S}_j^{\left(n_{m-2}^{\prime }\right)}\dots$ $\text{(}m$ factors). We have then that: $$\begin{array}{cc}\text{(££)}& U_r=U({\alpha }^{\vee \prime }\left(r\right),n_r^{\prime })\text{for some}{\alpha }^{\vee \prime }\left(r\right)\in R_{\{i,j\}}^{\vee }\end{array}$$ (In fact we have ${\alpha }^{\vee \prime }\left(m\right)={\alpha }_j^{\vee },$ ${\alpha }^{\vee \prime }(m-1)={\sigma }_j\left({\alpha }_i^{\vee }\right),$ ${\alpha }^{\vee \prime }(m-2)={\sigma }_j{\sigma }_i\left({\alpha }_j^{\vee }\right),$ etc.) To be able to conclude from (£) and (££) that $n_r=n_r^{\prime }$ we have to rule out the possibility that ${\alpha }^{\vee \prime }\left(r\right)=-{\alpha }^{\vee }\left(r\right)\text{.}$ For this consider the galleries of the simple hyperplane system $(𝕍,\stackrel{\circ }{I},ℳ)\text{.}$ We know that the signed gallery $$C^{+}{C_1}^{+}{C}_2\dots {}^{+}{C}_m=C^{U_1^{\prime }}{C_1}^{U_2^{\prime }}{C}_2\dots {}^{U_m^{\prime }}{C}_m$$ is flippable and its flipped form is $$C^{+}{D_1}^{+}{D}_2\dots {}^{+}{D}_m=C^{U_m^{\prime }}{D_1}^{U_{m-1}^{\prime }}{D}_2\dots {}^{U_1^{\prime }}{D}_m$$ where $U_r^{\prime }=U({\alpha }^{\vee }\left(r\right),+)=U\left({\alpha }^{\vee \prime }(r,+)\right)$ (definition (1.6)). It follows that ${\alpha }^{\vee \prime }\left(r\right)$ and ${\alpha }^{\vee }\left(r\right)$ are the same (positive) dual roots. (iii) We know that $U\prime ≔U_1^{\prime }\cap \dots \cap U_m^{\prime }$ is nonempty so pick an $x\in U\prime \text{.}$ Then for $\lambda >0$ small enough we have ${\alpha }^{\vee }\left(r\right)$ $\left(\lambda x\right)<1$ for $r=1,\dots ,m,$ so $\lambda x\in \bigcap _{r=1}^{m}U({\alpha }^{\vee }\left(r\right),0)≕U_0,$ hence (iii). (iv) Is obvious. (v) If $G$ is flippable, let $i,j\in \{1,\dots ,\ell \}$ be as in (i) and $U_0$ as in the proof of (iii). Note that $0\in {\bar{U}}_0\text{.}$ Set $J≔\{i,j\}\text{.}$ Then $W_J$ is finite, so ${\stackrel{\sim }{W}}_J$ is an affine group, acting transitively on the chambers w.r.t. its mirrors (see (1.5)). Since $ℳ(C,C_m)={ℳ}_J,$ $U$ is such a chamber and so is $U_0\text{.}$ So there exists a unique $w·e^q\in {\stackrel{\sim }{W}}_J,$ $w\in W_J,$ $q\in Q_J$ such that $w·e^q\left(U\right)=U_0\text{.}$ Then $0\in w·e^q\left(\bar{U}\right)$ and because $w\left(0\right)=0$ we also have $0\in q+\bar{U}\text{.}$ $\square$

(3.8) Let $i,j\in \{1,\dots ,\ell \}$ and $m≔m_{ij}<\infty \text{.}$ We want to determine explicitly the set $N_{ij}\subset {\mathbb{Z}}^m$ of elements $(n_1,\dots ,n_m)$ such that (3.6) is flippable. Since $N_{ji}=\{(n_1,\dots ,n_m)\in {\mathbb{Z}}^m\hspace{0.17em}|\hspace{0.17em}(n_m,\dots ,n_1)\in N_{ij}\}$ by (3.5)(ii) we can restrict ourselves to $n_{ij}\le n_{ji},$ so, since $n_{ij}{n}_{ji}<4,$ there are four cases: $(n_{ij},n_{ji})=(0,0),(-1,-1),(-1,-2)$ and $(-1,-3)$ respectively. The positive dual roots of ${\alpha }^{\vee }\left(r\right)\in R_{ij}^{\vee },$ $r=1,\dots ,m$ are linear combinations $x\left(r\right){\alpha }_i^{\vee }+y\left(r\right){\alpha }_j^{\vee }$ with integer coefficients $x\left(r\right)$ and $y\left(r\right)$ given by the following table:

$$\begin{array}{|ccccc|}\hline \text{CASE}& \text{1.}& \text{2.}& \text{3.}& \text{4.}\\ m& 2& 3& 4& 6\\ (n_{ij},n_{ji})& (0,0)& (-1,-1)& (-1,-2)& (-1,-3)\\ r& 1\hspace{0.17em}2& 1\hspace{0.17em}2\hspace{0.17em}3& 1\hspace{0.17em}2\hspace{0.17em}3\hspace{0.17em}4& 1\hspace{0.17em}2\hspace{0.17em}3\hspace{0.17em}4\hspace{0.17em}5\hspace{0.17em}6\\ x\left(r\right)& 1\hspace{0.17em}0& 1\hspace{0.17em}1\hspace{0.17em}0& 1\hspace{0.17em}2\hspace{0.17em}1\hspace{0.17em}0& 1\hspace{0.17em}3\hspace{0.17em}2\hspace{0.17em}3\hspace{0.17em}1\hspace{0.17em}0\\ y\left(r\right)& 0\hspace{0.17em}1& 0\hspace{0.17em}1\hspace{0.17em}1& 0\hspace{0.17em}1\hspace{0.17em}1\hspace{0.17em}1& 0\hspace{0.17em}1\hspace{0.17em}1\hspace{0.17em}2\hspace{0.17em}1\hspace{0.17em}1\\ \hline\end{array}$$

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

Case 3. $(n_{ij},n_{ji})=(-1,-2),$ then

$${\alpha }^{\vee }\left(2\right)={\sigma }_i\left({\alpha }_j^{\vee }\right)={\alpha }_j^{\vee }-{\alpha }_j^{\vee }\left({\alpha }_i\right){\alpha }_i^{\vee }={\alpha }_j^{\vee }-n_{ji}{\alpha }_i^{\vee }=2{\alpha }_i^{\vee }+{\alpha }_j^{\vee }\text{.}$$

Let $U$ be as in (3.5)(i). If $z\in U,$ then the numbers $a≔{\alpha }_i^{\vee }\left(z\right)$ and $b≔{\alpha }_j^{\vee }\left(z\right)$ satisfy, since ${\alpha }^{\vee }\left(r\right)\left(z\right)=x\left(r\right)a+y\left(r\right)b$ and $z\in U_r=U({\alpha }^{\vee }\left(r\right),n_r),$ the following:

$$\begin{array}{cc}\text{(\$)}& n_r<x\left(r\right)a+y\left(r\right)b<n_r+1r=1,\dots ,m\text{.}\end{array}$$

Conversely, since ${\alpha }_i^{\vee }$ and ${\alpha }_j^{\vee }$ are linearly independent the existence of numbers $a$ and $b$ such that ($) hold, guarantees that $U\ne \varnothing \text{.}$ Now we look for necessary and sufficient conditions on $(n_1,\dots ,n_m)$ for the existence of $a$ and $b$ with ($) in each special case:

Case 1. $n_{ij}=n_{ji}=0,$ $m=2\text{.}$

$$\begin{array}{cc}{n}_1<a<n_1+1& \text{No conditions on}\hspace{0.17em}(n_1,n_2)\\ n_2<b<n_2+1& \text{so}\hspace{0.17em}{N}_{ij}={\mathbb{Z}}^2\end{array}$$

Case 2. $n_{ij}=n_{ji}=-1,$ $m=3\text{.}$

(1)	$n_1<a<n_1+1$
(2)	$n_2<a+b<n_2+1$
(3)	$n_3<b<n_3+1$

(1) and (3) imply $n_1+n_3<a+b<n_1+n_3+2,$ so with (2) we see that $n_2=n_1+n_3$ or $n_2=n_1+n_3+1,$ so $n_1-n_2+n_3\in \{0,-1\}\text{.}$ And conversely if the latter is satisfied, there are $a,b\in ℝ$ such that (1), (2) and (3) hold. So $N_{ij}=\{(n_1,n_2,n_3)\in {\mathbb{Z}}^3\hspace{0.17em}|\hspace{0.17em}{n}_1-n_2+n_3\in \{0,-1\}\}\text{.}$

$$\begin{array}{c} 0 -1 n2+1 n2 n1 n1+1 \\ \text{Figure 19.}\end{array}$$

Case 3. $n_{ij}=-1$ and $n_{ji}=-2,$ $m=4\text{.}$

(1)	$n_1<a<n_1+1$
(2)	$n_2<2a+b<n_2+1$
(3)	$n_3<a+b<n_3+1$
(4)	$n_4<b<n_4+1$

From (1), (3) and (4) it follows that $n_1-n_3+n_4\in \{0,-1\}$ and from (1), (2) and (3) it follows that $n_1-n_2+n_3\in \{0,-1\}\text{.}$ So $(n_1-n_3+n_4,n_1-n_2+n_3)\in {\{0,-1\}}^2$ and conversely in each of the four possibilities, there are $a,b\in ℝ$ satisfying (1)-(4). The point $(a,b)$ can be found in the region indicated in figure 20. Hence $N_{ij}=\{(n_1,\dots ,n_4)\in {\mathbb{Z}}^4\hspace{0.17em}|\hspace{0.17em}(n_1-n_3+n_4,n_1-n_2+n_3)\in {\{0,-1\}}^2\}\text{.}$

$$\begin{array}{c} 0 0 0 -1 -1 0 -1 -1 n4+1 n4 n1 n1+1 \\ \text{Figure 20.}\end{array}$$

Case 4. $n_{ij}=-1$ and $n_{ji}=-3,$ $m=6\text{.}$

(1)	$n_1<a<n_1+1$
(2)	$n_2<3a+b<n_2+1$
(3)	$n_3<2a+b<n_3+1$
(4)	$n_4<3a+2b<n_4+1$
(5)	$n_5<a+b<n_5+1$
(6)	$n_6<b<n_6+1$

$$\begin{array}{c} 0 0 0 0 0 0 0 -1 0 0 -1 0 0 -1 0 0 0 -1 0 -1 0 -1 -1 0 -1 0 0 -1 -1 0 -1 0 -1 0 -1 -1 -1 -1 0 -1 -1 -1 -1 0 -1 -1 -1 -1 n6+1 n6 n1 n1+1 \\ \text{Figure 21.}\end{array}$$

As in the previous cases it can be derived that $k≔(k_1,k_2,k_3,k_4)\in {\{0,-1\}}^4$ where $k$ is given by $k≔(n_1-n_5+n_6,n_1-n_3+n_5,n_1-n_2+n_3,n_2-n_4+n_6)\text{.}$ But not all 16 possibilities occur for we have an extra condition, for instance $k_1\ne k_3\Rightarrow k_4=k_1\text{.}$ In fact it can be checked that we have the follow: There exist $a,b\in ℝ$ satisfying (1)-(6) if and only if $k\in A,$ where $A≔\{(k_1,k_2,k_3,k_4)\in {\{0,-1\}}^4\hspace{0.17em}|\hspace{0.17em}{k}_1\ne k_3\Rightarrow k_4=k_1\}\text{.}$ The 12 open regions in the open square $(n_1,n_1+1)\times (n_6,n_6+1)$ where $(a,b)$ can be found corresponding to the 12 possibilities of $k\in A$ are depicted in Figure 21.

(3.9) Definition. For $i\in \{1,\dots ,\ell \}$ and $n\in \mathbb{Z}$ we define the elements

$${\hat{s}}_i^{\left(n\right)}≔\nu \left(S_i^{\left(n\right)}\right)\in {\pi }^{-1}\left(W\right)(\nu :\text{Gal}/W\to {\pi }^{-1}\left(W\right)\text{as in (3.1)})$$

It can be checked that $s_i={\hat{s}}_i^{\left(0\right)},$ where $s_i\in \pi (X,*)$ is as defined in (2.4).

(3.10) Corollary of lemma (3.3). ${\left({\hat{s}}_i^{\left(n\right)}\right)}^{-1}={\hat{s}}_i^{(-n-1)}\text{.}$

(3.11) Corollary $(n=0)\text{.}$ $s_i^{-1}={\hat{s}}_i^{(-1)}$ (But in general $s_i^n\ne {\hat{s}}_i^{\left(n\right)}$ ??)

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

(3.12) Theorem. The group ${\pi }^{-1}\left(W\right)$ is generated by ${\hat{s}}_i^{\left(n\right)},$ $i\in \{1,\dots ,\ell \},$ $n\in \mathbb{Z}\text{.}$ A complete set of relations is:

(1)

${\hat{s}}_i^{\left(n\right)}{\hat{s}}_i^{(-n-1)}$ for $i\in \{1,\dots ,\ell \},$ $n\in \mathbb{Z}\text{.}$

(2)

${\hat{s}}_i^{\left(n_1\right)}{\hat{s}}_j^{\left(n_2\right)}{\hat{s}}_i^{\left(n_3\right)}\dots ={\hat{s}}_j^{\left(n_m\right)}{\hat{s}}_i^{\left(n_{m-1}\right)}{\hat{s}}_j^{\left(n_{m-2}\right)}\dots$ where $m≔m_{ij}<\infty$ and both sides of (2) have $m$ factors and $(n_1,\dots ,n_m)\in N_{ij}\text{.}$ The sets $N_{ij}\in {\mathbb{Z}}^m$ are determined as follows: $$\begin{array}{c}{n}_{ij}=n_{jj}=0\Rightarrow N_{ij}={\mathbb{Z}}^2\\ n_{ij}=n_{jj}=-1\Rightarrow N_{ij}=\{(n_1,n_2,n_3)\in {\mathbb{Z}}^3\hspace{0.17em}|\hspace{0.17em}{n}_1-n_2+n_3\in \{0,-1\}\}\\ n_{ij}=-1,n_{jj}=-2\Rightarrow N_{ij}=\{(n_1,n_2,n_3,n_4)\in {\mathbb{Z}}^4\hspace{0.17em}|\hspace{0.17em}(n_1-n_3+n_4,n_1-n_2+n_3)\in {\{0,-1\}}^2\}\\ n_{ij}=-1,n_{jj}=-3\Rightarrow N_{ij}=\{(n_1,\dots ,n_6)\in {\mathbb{Z}}^6\hspace{0.17em}|\hspace{0.17em}(n_1-n_5+n_6,n_1-n_3+n_5,n_1-n_2+n_3,n_2-n_4+n_6)\in A\}\end{array}$$ where $A≔\{(k_1,k_2,k_3,k_4)\in {\{0,-1\}}^4\hspace{0.17em}|\hspace{0.17em}{k}_1\ne k_3\Rightarrow k_4=k_1\}\text{.}$

		Proof.
	We have the surjective homomorphism of semigroups $\nu :\text{Gal}/W\to {\pi }^{-1}\left(W\right)$ (3.1). So the theorem follows from the fact that $\text{Gal}/W$ is freely generated by the elements $S_i^{\left(n\right)}$ and the remark that two $W\text{-orbits}$ of galleries have the same $\nu \text{-image,}$ if and only if they are equivalent by means of the $W\text{-orbits}$ 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). $\square$

(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 ${\pi }^{-1}\left(W\right)$ 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 $A_W$ and $Q$ in $\pi (X,*)$ (lemma (4.1) and (4.3)). As long as the proof is not finished, it is important to distinguish between the fundamental group $\pi (X,*)$ and the abstract group $A_{\stackrel{\sim }{W}},$ which can be described as follows (cf. def.(2.13) and cor.(2.10)):

Add to the free product of $A_W$ and $Q$ the push relations (2.6). The inbeddings $A_W\to \pi (X,*),$ $Q\to \pi (X,*)$ induce a homomorphism $A_W*Q\to \pi (X,*),$ which yields, after we have established the push relations (corollary (4.12)), a homomorphism of the quotient $A_{\stackrel{\sim }{W}}$ of $A_W*Q$ to $\pi (X,*)\text{.}$ 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 ${\pi }^{-1}\left(W\right)*Q:A_{\stackrel{\sim }{W}}\stackrel{f}{\to }G\stackrel{h}{\to }\pi (X,*)\text{.}$ 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 $s_1,\dots ,s_{\ell }$ as defined in (2.4) and the subgroup of ${\pi }^{-1}\left(W\right)$ generated by them form an Artin system. In particular $A_W$ is inbedded in ${\pi }^{-1}\left(W\right)\text{.}$

		Proof.
	It follows from proposition (3.5)(ii) and (iii) that the elements $s_i\in \pi (X,*)$ satisfy the relations $s_i{s}_j{s}_i\dots =s_j{s}_i{s}_j\dots$ $i\ne j,$ $m_{ij}<\infty$ and $m_{ij}$ factors on both sides. So there is a homomorphism $A_W\to \pi (X,*)\text{.}$ The image is inside ${\pi }^{-1}\left(W\right)\text{.}$ The fact that $Y\subset Y(𝕍,\stackrel{\circ }{I},ℳ)$ induces a homomorphism $\pi (Y/W)\cong {\pi }^{-1}\left(W\right)\to \pi (Y(𝕍,\stackrel{\circ }{I},ℳ)/W)=A_W,$ which is a left inverse, so the first homomorphism was injective. $\square$

(4.2) Definition. If $q\in Q,$ let $T_q$ be the curve $[0,1]\ni t\to tq+\sqrt{-1}c\in Y\text{.}$ Define $e^q≔\{p\circ T_q\}\in \pi (X,*)\text{.}$

Note that (see (2.4)) ${𝒯}_i=T_{{\alpha }_i},$ hence $t_i=e^{{\alpha }_i}\text{.}$ In view of remark (1.2) the use of the notation $e^q$ needs justification, which is given by the following lemma:

(4.3) Lemma. The map $Q\ni q\to e^q\in \pi (X,*)$ is an injective homomorphism.

		Proof.
	It is easy to see that the composition with $\pi :\pi (X,*)$ induces the identity on $Q\text{.}$ $\square$

Now we prove a prenatal form of the push relations.

(4.4) Proposition. In $\pi (X,*)$ we have the following relations:

$$\begin{array}{cc}\text{(4.5)}& {\hat{s}}_i^{\left(n\right)}·e^q=e^{{\sigma }_i\left(q\right)}·{\hat{s}}_i^{(n+{\alpha }_i^{\vee }\left(q\right))}\text{for}\hspace{0.17em}i\in \{1,\dots ,\ell \},n\in \mathbb{Z},q\in Q\text{.}\end{array}$$

		Proof.
	Equivalently: (see figure 18 for this proof) $$\begin{array}{cc}\text{(4.6)}& $ e^{{\sigma }_i(-q)}$·\nu \left(S_i^{\left(n\right)}\right)·e^q=\nu \left(S_i^{(n+{\alpha }_i^{\vee }\left(q\right))}\right)\end{array}$$ Let $S$ be the curve in $Y$ defined by $$S\left(t\right)≔\rho \left(t\right)+\sqrt{-1}[(1-t)c+t{\sigma }_i\left(c\right)],\text{where}\hspace{0.17em}\rho :[0,1]\to 𝕍\hspace{0.17em}\text{is continuous}$$ with the property that $n<{\alpha }_i^{\vee }\left(\rho \left(\frac{1}{2}\right)\right)<n+1$ and $\rho \left(0\right)=\rho \left(1\right)=0\text{.}$ Then $\phi \left(C^{U({\alpha }_i^{\vee },n)}{\sigma }_i\left(C\right)\right)=\left\{S\right\},$ so $p\circ S$ is a curve in $X$ representing $\nu \left(S_i^{\left(n\right)}\right)\text{.}$ Define curves ${\gamma }_1,{\gamma }_2,{\gamma }_3$ in $Y$ by $\text{(}t\in [0,1]\text{):}$ $$\begin{array}{cc}\text{(4.7)}& \begin{array}{c}{\gamma }_1\left(t\right)=T_q\left(t\right)=qt+\sqrt{-1}c\\ {\gamma }_2\left(t\right)=q+S\left(t\right)=q+\rho \left(t\right)+\sqrt{-1}[(1-t)c+t{\sigma }_i\left(c\right)]\\ {\gamma }_3\left(t\right)=q+{\sigma }_i\left(T_{{\sigma }_i(-q)}\right)=q-tq+\sqrt{-1}{\sigma }_i\left(c\right)\text{.}\end{array}\end{array}$$ Then $p\circ {\gamma }_j$ represent $e^q,$ $\nu \left(S_i^{\left(n\right)}\right)$ and $e^{{\sigma }_i(-q)}$ for $j=1,2$ and 3 respectively. Since $$\begin{array}{c}{\gamma }_1\left(1\right)={\gamma }_2\left(0\right)=q+\sqrt{-1}c\text{and}\\ {\gamma }_2\left(1\right)={\gamma }_3\left(0\right)=q+\sqrt{-1}{\sigma }_i\left(c\right)\end{array}$$ the curve $\gamma ={\gamma }_1{\gamma }_2{\gamma }_3$ is defined and $p\circ \gamma$ represents the left hand side of (4.6). So it suffices to prove that $$C^{U({\alpha }_i^{\vee },n+{\alpha }_i^{\vee }\left(q\right))}{\sigma }_i\left(C\right)\in L_{\gamma }\text{in the sense of def. (I.1.6).}$$ For this we check with the help of formulas (4.7) that the following hold: (i) ${\gamma }_1\left([0,1]\right)\cup {\gamma }_2\left([0,\frac{1}{2}]\right)\in Y_C$ (ii) ${\gamma }_2\left([\frac{1}{2},1]\right)\cup {\gamma }_3\left([0,1]\right)\in Y_{{\sigma }_i\left(C\right)}$ (iii) ${\gamma }_2\left(\frac{1}{2}\right)\in {\text{Re}}^{-1}\left(U({\alpha }_i^{\vee },n+{\alpha }_i^{\vee }\left(q\right))\right)$ Property (i) follows from $\text{Im}\left({\gamma }_1\left(t\right)\right)=c\in C$ and for $t\in [0,\frac{1}{2})$ we have $\text{Im}\left({\gamma }_2\left(t\right)\right)=(1-t)c+{\sigma }_i\left(c\right)\in C\text{.}$ Moreover for $t=\frac{1}{2}$ the latter is contained in $M_{{\alpha }_i^{\vee }}$ but then $\text{Re}\left({\gamma }_2\left(\frac{1}{2}\right)\right)=q+\rho \left(\frac{1}{2}\right)\notin M({\alpha }_i^{\vee },m)$ for all $m\in \mathbb{Z},$ because we have $n<{\alpha }_i^{\vee }\left(\rho \left(\frac{1}{2}\right)\right)<n+1$ so also: $$\begin{array}{cc}\text{(4.8)}& n+{\alpha }_i^{\vee }\left(q\right)<{\alpha }_i^{\vee }(q+\rho \left(\frac{1}{2}\right))<n+1+{\alpha }_i^{\vee }\left(q\right)\end{array}$$ so ${\alpha }_i^{\vee }(q+\rho \left(\frac{1}{2}\right))\notin \mathbb{Z}\text{.}$ This proves (i). (ii) is proved analogously and (iii) follows from (4.8). $\square$

(4.9) Definition. Let $G$ be the group which can be described as follows: Add to the free product of ${\pi }^{-1}\left(W\right)$ and $Q$ the relations (4.5).

The inclusion ${\pi }^{-1}\left(W\right)\subset \pi (X,*)$ and the inbedding $Q\to \pi (X,*)$ (4.3) induce a homomorphism ${\pi }^{-1}\left(W\right)*Q\to \pi (X,*)$ and by (4.4) this gives a homomorphism $h:G\to \pi (X,*)\text{.}$

(4.10) Proposition. $h$ is an isomorphism.

		Proof.
	$$\begin{array}{ccccccc}& & {\pi }^{-1}\left(W\right)*Q& \supset & {\pi }^{-1}\left(W\right)& \stackrel{\pi }{⟶}& W\\ & \swarrow & \downarrow & & & & \cap \\ G& \stackrel{h}{⟶}& \pi (X,*)& \multicolumn{3}{c}{\stackrel{\pi }{⟶}}& \stackrel{\sim }{W}& \supset & Q\end{array}$$ Since $\stackrel{\sim }{W}=W·Q,$ we have $\pi (X,*)={\pi }^{-1}\left(W\right)·Q$ so $h$ is surjective. By the relations (4.5), which by definition (4.9) hold in $G$ it follows that $G={\pi }^{-1}\left(W\right)·Q\text{.}$ Now let $x\in {\pi }^{-1}\left(W\right)$ and $q\in Q$ and consider $x·e^q\in G\text{.}$ Suppose $h(x·e^q)=1\text{.}$ Then $\pi \left(h\left(x\right)\right)·\pi \left(h\left(e^q\right)\right)$ is the unique decomposition of $\pi \left(h(x·e^q)\right)=\pi \left(1\right)=1$ in $\stackrel{\sim }{W},$ so $q=\pi \left(h\left(e^q\right)\right)=0\text{.}$ Hence $h\left(x\right)=1\text{.}$ Now $h$ is already bijective on ${\pi }^{-1}\left(W\right),$ as can be seen form the diagram. So $x=1\text{.}$ $\square$

Before we state the next proposition, we recall the definition of the elements $s_i^{\left(n\right)}$ for $i\in \{1,\dots ,\ell \}$ and $n\in \mathbb{Z}:$

$$\begin{array}{ccc}{s}_i^{\left(2m\right)}& ≔& t_i^m{s}_i{t}_i^m\\ s_i^{(2m-1)}& ≔& t_i^m{s}_i^{-1}{t}_i^m\end{array}$$

Since $s_i,s_i^{-1}\in A_W\subset {\pi }^{-1}\left(W\right)$ and $t_i\in Q$ these elements $s_i^{\left(n\right)}$ can be interpreted as elements of one of the groups $A_W*Q,$ $A_{\stackrel{\sim }{W}},$ ${\pi }^{-1}\left(W\right)*Q,$ $G$ and finally $\pi (X,*)\text{.}$ The elements ${\hat{s}}_i^{\left(n\right)}$ can be seen only as elements of the last three groups (unless $n=0$ or $-1\text{).}$ So the statement has only in this three cases a meaning. The statement is not true in ${\pi }^{-1}\left(W\right)*Q\text{.}$

(4.11) Proposition. In $G$ and $\pi (X,*)$ we have ${\hat{s}}_i^{\left(n\right)}=s_i^{\left(n\right)}$

		Proof.
	$\begin{array}{c}{\hat{s}}_i^{\left(2m\right)}={\hat{s}}_i^{\left({\alpha }_i^{\vee }\left(m{\alpha }_i\right)\right)}\underset{\text{(4.5)}}{=}{e}^{m{\alpha }_i}{\hat{s}}_i^{\left(0\right)}{e}^{m{\alpha }_i}\underset{\text{(3.9)}}{=}{t}_i^m{s}_i{t}_i^m≕s_i^{\left(2m\right)}\text{and}\\ {\hat{s}}_i^{(2m-1)}={\hat{s}}_i^{(-1+{\alpha }_i^{\vee }\left(m{\alpha }_i\right))}\underset{\text{(4.5)}}{=}{e}^{m{\alpha }_i}{\hat{s}}_i^{(-1)}{e}^{m{\alpha }_i}\underset{\text{(3.11)}}{=}{t}_i^m{s}_i^{-1}{t}_i^m≕s_i^{(2m-1)}\text{.}\end{array}$ $\square$

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

By definition the push relations (2.6) hold in $A_{\stackrel{\sim }{W}}\text{.}$ The fact that they hold in $G$ implies, that the inclusion $A_W*Q\subset {\pi }^{-1}\left(W\right)*Q$ induces a homomorphism $f:A_{\stackrel{\sim }{W}}\to G\text{.}$

(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 ${\pi }^{-1}\left(W\right)$ as follows: for $q\in Q,$ $x\in {\pi }^{-1}\left(W\right)$ and $\pi \left(x\right)≕w\in W$

$${\lambda }_q\left(x\right)≔e^{q}x{e}^{-w^{-1}\left(q\right)}$$

Note that ${\lambda }_q$ is not a homomorphism. Instead we have

$$\begin{array}{cc}\text{(4.15)}& {\lambda }_q(xy=){\lambda }_q\left(x\right){\lambda }_{w^{-1}\left(q\right)}\left(y\right),w≔\pi \left(x\right)\end{array}$$

In the same way we can define an action $\lambda$ of $Q$ on ${\underset{\_}{\pi }}^{-1}\left(W\right)$ as subgroup of $A_{\stackrel{\sim }{W}},$ where $\underset{\_}{\pi }:A_{\stackrel{\sim }{W}}\to \stackrel{\sim }{W}$ is the projection induced by the projection $A_W\to W\subset \stackrel{\sim }{W}$ and the identity $Q\to Q\stackrel{\sim }{W}\text{.}$ After the establishing of the theorem $\underset{\_}{\pi }$ and $\pi$ can be identified. This action of $Q$ corresponds to an action of $Q$ (shifting) on $\text{Gal}$ which is defined in the following lemma:

(4.16) Lemma. An action $\mu$ of $Q$ on $\text{Gal}$ is induced by ${\mu }_q\left(A^{U}B\right)≔A^{q+U}B\text{.}$ For $q\in Q$ and $w\in W$ we have:

$$\begin{array}{cc}\text{(4.17)}& w\circ {\mu }_q={\mu }_{w\left(q\right)}\circ w\end{array}$$

		Proof.
	Let $U\in K\left(p(A,B)\right)\text{.}$ We have to prove that $q+U\in K\left(p(A,B)\right)$ for $q\in Q\text{.}$ If $A=w\left(C\right)$ for $w\in W,$ then $U=w\left(U({\alpha }_i,n)\right)$ for some $n\in \mathbb{Z}$ and $i\in \{1,\dots ,\ell \}\text{.}$ Then $q+w\left(U({\alpha }_i,n)\right)=w(w^{-1}\left(q\right)+U({\alpha }_i,n))=\text{(by (1.7) v)}=w\left(U({\alpha }_i,n+{\alpha }_i^{\vee }\left(w^{-1}\left(q\right)\right))\right)\in K\left(p(A,B)\right)\text{.}$ Property (4.17) follows from $w\left(A^{q+U}B\right)={w\left(A\right)}^{w(q+U)}w\left(B\right)={w\left(A\right)}^{w\left(q\right)+w\left(U\right)}w\left(B\right)\text{.}$ $\square$

Since $\text{Gal}/W$ is freely generated by $S_i^{\left(n\right)}$ we have a homomorphism of semigroups $\rho :\text{Gal}/W\to A_{\stackrel{\sim }{W}}$ such that $\rho \left(S_i^{\left(n\right)}\right)=s_i^{\left(n\right)}\text{.}$ Denote the composition with $\text{Gal}\to \text{Gal}/W$ by $\beta ,$ so $\beta :\text{Gal}\to A_{\stackrel{\sim }{W}}\text{.}$

(4.18) Lemma. the functor $\beta$ is equivariant w.r.t the action $\lambda$ and $\mu$ restricted to ${\text{Gal}}_C,$ i.e. for $G\in \text{Gal}$ and $\text{beg}\left(G\right)=C,$ $q\in Q$ we have:

$${\lambda }_q\left(\beta \left(G\right)\right)=\beta \left({\mu }_q\left(G\right)\right)\text{.}$$

		Proof.
	First we prove: Suppose the lemma is true for $G$ and $G\prime$ and let $\text{end}\left(G\right)=w\left(C\right),$ $w\in W\text{.}$ Then the lemma is true for $G·w\left(G\prime \right)\text{.}$ $$\begin{array}{ccc}{\lambda }_q\left(\beta (G·w\left(G\prime \right))\right)& =& {\lambda }_q(\beta \left(G\right)·\beta \left(G\prime \right))\\ & \underset{\text{(4.15)}}{=}& {\lambda }_q\left(\beta \left(G\right)\right)·{\lambda }_{w^{-1}\left(q\right)}\left(\beta \left(G\prime \right)\right)\\ & =& \beta \left({\mu }_q\left(G\right)\right)·\beta \left({\mu }_{w^{-1}\left(q\right)}\left(G\prime \right)\right)\\ & \underset{\text{(4.17)}}{=}& \beta \left({\mu }_q\left(G\right)\right)·\beta \left({\mu }_q\left(w\left(G\prime \right)\right)\right)\\ & =& \beta \left({\mu }_q(G·w\left(G\prime \right))\right)\text{.}\end{array}$$ So we are done if we check the lemma for a one step gallery: $$\begin{array}{ccc}\beta \left({\mu }_q\left(C^{U({\alpha }_i,n)}{\sigma }_i\left(C\right)\right)\right)& =& \beta \left(C^{q+U({\alpha }_i,n)}{\sigma }_i\left(C\right)\right)\\ & =& \beta \left(C^{U({\alpha }_i,n+{\alpha }_i^{\vee }\left(q\right))}{\sigma }_i\left(C\right)\right)\\ & =& \rho \left(s_i^{(n+{\alpha }_i^{\vee }\left(q\right))}\right)\\ & =& s_i^{(n+{\alpha }_i^{\vee }\left(q\right))}\end{array}$$ On the other hand: $$\begin{array}{ccc}{\lambda }_q\left(\beta \left(C^{U({\alpha }_i,n)}{\sigma }_i\left(C\right)\right)\right)& =& {\lambda }_q\left(\rho \left(S_i^{\left(n\right)}\right)\right)\\ & =& {\lambda }_q\left(s_i^{\left(n\right)}\right)\\ & =& e^q{s}_i^{\left(n\right)}{e}^{-{\sigma }_i\left(q\right)}\end{array}$$ and by (2.9) this equals also $s_i^{(n+{\alpha }_i^{\vee }\left(q\right))}\text{.}$ $\square$

(4.19) Proposition. There exists a unique homomorphism $g:{\pi }^{-1}\left(W\right)\to A_{\stackrel{\sim }{W}}$ such that we have $\rho =g\circ \nu \text{.}$ Note that $g\left({\hat{s}}_i^{\left(n\right)}\right)=s_i^{\left(n\right)}\text{.}$

$$\begin{array}{ccc}& & \text{Gal}/W\\ & \nu \swarrow & & \searrow \rho \\ {\pi }^{-1}\left(W\right)& & \underset{g}{⟶}& & A_{\stackrel{\sim }{W}}\end{array}$$

		Proof.
	It is enough to check that the elements $s_i^{\left(n\right)}\in A_{\stackrel{\sim }{W}}$ satisfy the cancel relations and the flip relations. First the cancel relations: $$\begin{array}{c}{s}_i^{\left(2m\right)}{s}_i^{(-2-1)}=t_i^m{s}_i{t}_i^m{t}_i^{-m}{s}_i^{-1}{t}_i^{-m}=1\text{and}\\ s_i^{(2m-1)}{s}_i^{(-2m+1-1)}=t_i^m{s}_i^{-1}{t}_i^m{t}_i^{-m}{s}_i{t}_i^{-m}=1\end{array}$$ Now the flip relations: Suppose $i,j\in \{1,\dots ,\ell \}$ and $m≔m_{ij}<\infty$ and $(n_1,\dots ,n_m)\in N_{ij}\text{.}$ Let $x≔s_i^{\left(n_1\right)}{s}_i^{\left(n_2\right)}{s}_i^{\left(n_3\right)}\dots$ and $y≔s_j^{\left(n_m\right)}{s}_i^{\left(n_{m-1}\right)}\dots$ (both $m$ factors). We have to prove: $x=y\text{.}$ Let $q\in Q$ be as exists by (3.5)(v), i.e. $0\in q+\bar{U}\text{.}$ Then it follows of lemma (4.18) and proposition (3.15)(iv) that ${\lambda }_q\left(x\right),$ ${\lambda }_q\left(y\right)$ ? $A_W\text{.}$ Since $f\left(x\right)=f\left(y\right),$ because the flip relations hold by definition in $G,$ we have ${\lambda }_q\left(f\left(x\right)\right)={\lambda }_q\left(f\left(y\right)\right),$ hence also $f\left({\lambda }_q\left(x\right)\right)=f\left({\lambda }_q\left(y\right)\right)\text{.}$ But $f$ is injective on $A_W$ as follows from the diagram, so ${\lambda }_q\left(x\right)={\lambda }_q\left(y\right)\text{.}$ Hence $x=y\text{.}$ $$\begin{array}{c}{A}_W\\ \cap & & \multicolumn{2}{c}{ }\\ A_W*Q\\ \downarrow \\ A_W& \stackrel{f}{⟶}& G& \stackrel{h}{⟶}& \pi (X,*)\end{array}$$ $\square$

		Proof of proposition (4.13).
	The homomorphism $g:{\pi }^{-1}\left(W\right)\to A_{\stackrel{\sim }{W}}$ induces a homomorphism ${\pi }^{-1}\left(W\right)*Q\to A_{\stackrel{\sim }{W}}$ and this induces a homomorphism $G\to A_{\stackrel{\sim }{W}}$ which is an inverse of $f:A_{\stackrel{\sim }{W}}\to G$ as easily can be checked on various generators. $\square$

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 $W_a$ of affine transformations of an affine space $E,$ such that $W_a$ 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 $\xi \in {ℂ}^{*}\text{),}$ which we will encounter in (5.3). One possibility (corresponding to $\xi =1+\sqrt{-1}\text{)}$ is obvious by the affine nature of $E:$ Let $E_{ℂ}≔E\times E$ and let the group $W_a$ act on $E_{ℂ}$ by acting on both components. The complexified reflection hyperplanes are then of the form $M+\sqrt{-1}M,$ where $M$ is a real affine reflection hyperplane of $W_a\text{.}$ 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 $W_a$ be a linear Coxeter group acting on a real vector space $𝕍\prime \text{.}$ $W_a$ is called an affine (or parabolic) linear Coxeter group is there is an invariant hyperplane $𝕍\subset 𝕍\prime ,$ with an invariant innerproduct and $𝕍\cap I=\left\{0\right\}\text{.}$

Let $\psi \in {\left(𝕍\prime \right)}^{*}$ such that $𝕍={\psi }^{-1}\left(0\right)$ and $\stackrel{\circ }{I}\subset {\psi }^{-1}\left((0,\infty )\right)\text{.}$ Then $E≔{\psi }^{-1}\left(1\right)$ is also invariant and $W_a$ acts on $E$ as an affine Coxeter group (in particular $\stackrel{\circ }{I}={\psi }^{-1}\left((0,\infty )\right)\text{)}$ as described in def. (5.1). Conversely given an affine Coxeter group as in (5.1) we can form $𝕍\prime ≔E\times ℝ,$ considering $E$ as a vector space. Now each (affine) reflection in $E$ extends uniquely to a linear reflection in $𝕍\prime$ and in this way we get a parabolic linear Coxeter group. The form $\psi$ can be defined by $\psi (x,t)≔t\text{.}$

(5.3) Complexification. Let $W_a$ be an affine linear Coxeter group in a space $𝕍\prime$ and let $\psi \in {\left(𝕍\prime \right)}^{*}$ be as before. Let $Y\prime$ and $X\prime$ be the space of regular points and the regular orbit space respectively of $W_a$ as defined in (II.3.2) and (II.3.3).

Let $\xi \in {ℂ}^{*}$ and consider $E_{ℂ}^{\xi }≔{\psi }^{-1}\left(\xi \right)\text{.}$ Let $Y_{\xi }≔E_{ℂ}^{\xi }-\bigcup _{M\in ℳ}{M}_{ℂ}$ and $X_{\xi }≔Y_{\xi }/W_a\text{.}$ The map $Y_1\ni z\to \xi z\in Y_{\xi }$ is a homeomorphism equivariant under the action of $W_a,$ so $X_{\xi }$ is homeomorphic to $X_1\text{.}$ On the other hand for $\text{Im}\left(\xi \right)>0,$ we have that $Y_{\xi }$ is a deformation retract of $Y\prime ,$ because if $z\in Y\prime ,$ then $z_{\xi }≔\frac{\xi }{\psi \left(z\right)}·z\in Y_{\xi }$ and $[z,z_{\xi }]\subset Y\prime \text{.}$ This deformation retraction is also equivariant for the action of $W_a,$ so $X_{\xi }$ is a deformation retract of $X\prime \text{.}$

(5.4) In particular this shows that the result of Nguyen Viet Dung, which in fact says that the fundamental group of $X_{1+\sqrt{-1}}$ is the Artin group associated to $W_a,$ is a special case of theorem (II.3.8).

We apply these ideas to the situation of an extended Coxeter group $\stackrel{\sim }{W}$ 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 ${\alpha }_0^{\vee }$ be the longest dual root, ${\alpha }_0$ the corresponding root. The corresponding reflection ${\sigma }_{{\alpha }_0}\in W$ is defined by for $v\in 𝕍:$ ${\sigma }_{{\alpha }_0}\left(v\right)≔v-{\alpha }_0^{\vee }\left(v\right){\alpha }_0\text{.}$ Define also the affine reflection ${\sigma }_0\in \stackrel{\sim }{W}$ by ${\sigma }_0≔e^{{\alpha }_0}{\sigma }_{{\alpha }_0}\text{.}$ Let $W_a$ be the group of affine transformations of $𝕍$ generated by $\{{\sigma }_0,{\sigma }_1,\dots ,{\sigma }_{\ell }\}\text{.}$ Then it is known (Bourbakl [Bou1968] Ch.VI §2) that $(W_a,\{{\sigma }_0,{\sigma }_1,\dots ,{\sigma }_{\ell }\})$ is a (parabolic) Coxeter system, $\bar{C}\cap \{x\in 𝕍\hspace{0.17em}|\hspace{0.17em}{\alpha }_0^{\vee }\left(x\right)\le 1\}$ is a fundamental domain of $W_a$ and $W_a=\stackrel{\sim }{W}\text{.}$ We want to establish an isomorphism between the (parabolic) Artin group $A_{W_a}$ associated to $W_a$ and the extended Artin group $A_{\stackrel{\sim }{W}}:$

(5.5) Theorem. Let $B=\{{\alpha }_1,\dots ,{\alpha }_{\ell }\},$ ${}^{\vee }:B\to B^{\vee }$ be the root basis of a finite irreducible root system. Let ${\alpha }_0^{\vee }$ be the longest dual root, ${\alpha }_0$ the corresponding root, ${\sigma }_{{\alpha }_0}$ the corresponding reflection in the Weyl group $$ W\text{.}$$

Let $(A_{W_a},\{{\bar{s}}_0,{\bar{s}}_1,\dots ,{\bar{s}}_{\ell }\})$ be the Artin system associated to the Coxeter system $(W_a,\{{\sigma }_0,{\sigma }_1,\dots ,{\sigma }_{\ell }\}),$ where ${\sigma }_0≔e^{{\alpha }_0}{\sigma }_{{\alpha }_0}\in \stackrel{\sim }{W}\text{.}$

Then $A_{W_a}$ and $A_{\stackrel{\sim }{W}}$ are isomorphic and an isomorphism is induced by:

$$\begin{array}{cc}\text{(5.6)}& \begin{array}{cccc}{\bar{s}}_i& ⟼& s_i& \text{for}i=1,\dots ,\ell \\ {\bar{s}}_0& ⟼& t_0{s}_{{\alpha }_0}^{-1}\end{array}\end{array}$$

where $t_0≔e^{{\alpha }_0}$ and $s_{{\alpha }_0}$ is the image of ${\sigma }_{{\alpha }_0}$ under the injection $W\to A_W$ discussed in (II.3.?).

		Proof.
	From the fact that $W$ is finite it follows that $\stackrel{\sim }{W}$ is an affine Coxeter group acting on $𝕍$ as discussed in the beginning of this paragraph. We construct the (affine) linear Coxeter system $(W_a,\{{\sigma }_0^{\prime },{\sigma }_1^{\prime },\dots ,{\sigma }_{\ell }^{\prime }\})$ as suggested there. Put $$\begin{array}{ccc}𝕍\prime & ≔& 𝕍\times ℝ\\ v_i& ≔& ({\alpha }_i,0),{\varphi }_i(v,t)≔{\alpha }_i^{\vee }\left(v\right)i=1,\dots ,\ell \\ v_0& ≔& (-{\alpha }_0,0),{\varphi }_0(v,t)≔t-{\alpha }_0^{\vee }\left(v\right)\end{array}$$ And as usual: ${\sigma }_i^{\prime }(v,t)≔(v,t)-{\varphi }_i(v,t)v_i$ for $i=1,\dots ,\ell \text{.}$ Now $W_a$ leaves invariant $E_{ℂ}^{\xi }≔{𝕍}_C\times \left\{\xi \right\}\subset {𝕍}_{ℂ}\times ℂ={\left(𝕍\prime \right)}_{ℂ}$ for each $\xi \in {ℂ}^{*}\text{.}$ Let us in particular study this action for $\xi =1\text{.}$ For $i=1,\dots ,\ell$ we have: $$\begin{array}{ccc}{\sigma }_i^{\prime }(v,1)& =& (v,1)-{\alpha }_i^{\vee }(v,1)v_i\\ & =& (v,1)-{\alpha }_i^{\vee }\left(v\right)({\alpha }_i,0)\\ & =& ({\sigma }_i\left(v\right),1)\\ {\sigma }_0^{\prime }(v,1)& =& (v,1)-{\alpha }_0^{\vee }(v,1)v_0\\ & =& (v,1)-(1-{\alpha }_0^{\vee }\left(v\right))(-{\alpha }_0,0)\\ & =& (v-{\alpha }_0^{\vee }\left(v\right){\alpha }_0+{\alpha }_0,1)\\ & =& ({\sigma }_{{\alpha }_0}\left(v\right)+{\alpha }_0,1)\\ & =& ({\sigma }_0\left(v\right),1)\end{array}$$ So the action of $W_a$ on ${𝕍}_{ℂ}\times \left\{1\right\}\cong {𝕍}_{ℂ}$ corresponds with the action of $\stackrel{\sim }{W}$ on ${𝕍}_{ℂ}\text{.}$ Hence $X\cong X_1\text{.}$ So by the reasoning in (5.3) we have $X\prime \simeq X,$ hence $A_{W_a}\cong A_{\stackrel{\sim }{W}}\text{.}$ We check that an isomorphism is explicitly given by (5.6). Consider the diagram $$\begin{array}{cccccccc}& Y\prime & \multicolumn{5}{c}{\underset{g}{⟶}}& Y\\ p\prime & \downarrow & {}_{\cong }\searrow ^{\supset }& \multicolumn{3}{c}{}& {\nearrow }_{\cong }& \downarrow & p\\ & Y_{\sqrt{-1}}& \underset{\text{(\$\$)}}{\overset{\cong }{⟶}}& Y_{1\phantom{\sqrt{1}}}\\ \phantom{\nearrow }\\ & X\prime & \multicolumn{5}{c}{\underset{f}{⟶}}& X\end{array}$$ where ($) is the deformation retraction defined in (5.3) and ($$) is the multiplication with $-\sqrt{-1}\text{.}$ The basepoint of $X\prime$ is $p\prime \left(\sqrt{-1}(c,1)\right),$ its image under $f:f\circ p\prime \left(\sqrt{-1}(c,1)\right)=p\left(g\left(\sqrt{-1}(c,1)\right)\right)=p\left(c\right)\text{.}$ But the basepoint of $X$ is $p\left(\sqrt{-1}c\right)\text{.}$ So we define ${\lambda }_1≔[\sqrt{-1}c,c]$ and then $p\circ {\lambda }_1$ is a curve from the basepoint of $X$ to the $f\text{-image}$ of the basepoint of $X\prime \text{.}$ So an explicit isomorphism $\pi (X\prime ,p\prime \left(\sqrt{-1}(c,1)\right))\to \pi (X,p\left(\sqrt{-1}c\right))$ is given by $$\begin{array}{cc}\text{(5.7)}& \{p\prime \circ \gamma \prime \}\to \{p\circ {\lambda }_1\}\{p\circ g\circ \gamma \prime \}{\{p\circ {\lambda }_1\}}^{-1}\end{array}$$ Now let $\gamma \prime$ be a curve in $Y\prime$ such that $p\prime \circ \gamma \prime$ represents the element ${\left({\bar{s}}_0\right)}^{-1},$ so for instance we can take $\gamma \prime \left(t\right)≔\rho \left(t\right)v_0+\sqrt{-1}\{(1-t)(c,1)+t{\sigma }_0(c,1)\}$ with $\rho \left(\frac{1}{2}\right)<0\text{.}$ Let $\gamma ≔g\circ \gamma \prime \text{.}$ Then $\gamma \left(t\right)=\sqrt{-1}\rho \left(t\right){\alpha }_0+(1-t)c+t{\sigma }_0\left(c\right)$ which is a curve from $c$ to ${\sigma }_0\left(c\right)={\sigma }_{{\alpha }_0}\left(c\right)+{\alpha }_0\text{.}$ Now we have to study the curve $\mu ={\lambda }_1\gamma {\lambda }_2^{-1},$ where ${\lambda }_2=[\sqrt{-1}{\sigma }_{{\alpha }_0}\left(c\right)+{\alpha }_0{\sigma }_{{\alpha }_0}\left(c\right)+{\alpha }_0]\text{.}$ For note that $p\circ {\lambda }_1=p\circ {\lambda }_2,$ so $p\circ \mu$ represents the right hand side of (5.7). The curve $\mu$ is illustrated in Figure 22, where the root system $B_2$ is shown. $$\begin{array}{cc} α0 α1 α2 σ0(c) c & c C σα0(c) \\ \text{Real part of}\hspace{0.17em}{\lambda }_1\gamma {\lambda }_2^{-1}& \text{Imaginary part of}\hspace{0.17em}{\lambda }_1\gamma {\lambda }_2^{-1}\\ \multicolumn{2}{c}{\text{Figure 22.}}\end{array}$$ Now we make four remarks about this curve $\mu \text{.}$ These remarks are sufficient to describe its homotopy type: (1) $\text{Im}\left(\mu \right)$ is a curve from $c$ to ${\sigma }_{{\alpha }_0}\left(c\right)\text{.}$ (2) $\text{Re}\left(\mu \right)$ is a curve from $0$ to ${\alpha }_0$ and except for the begin and end point there is only one other value of the parameter, say $t=\frac{1}{2}$ such that the real part is on some hyperplane $N\in p_M,$ $M\in ℳ\text{.}$ (3) We have $\text{Im}\left(\mu \left(\frac{1}{2}\right)\right)=\rho \left(\frac{1}{2}\right){\alpha }_0\text{.}$ Since $\rho \left(\frac{1}{2}\right)<0$ this means that the imaginary part has already passed by all the hyperplanes separating $C$ and ${\sigma }_{{\alpha }_0}\left(C\right)\text{.}$ (4) While $\text{Im}\left(\mu \right)$ is passing by the hyperplanes separating $C$ and ${\sigma }_{{\alpha }_0}\left(C\right)$ the real part is in $C\cap \{x\in 𝕍\hspace{0.17em}|\hspace{0.17em}{\alpha }_0^{\vee }\left(x\right)<1\}\text{.}$ From these considerations it follows that $p\circ \mu$ is a loop in $X$ representing the element $s_{{\alpha }_0}{e}^{-{\alpha }_0}\in A_{\stackrel{\sim }{W}}\text{.}$ So the image of ${\bar{s}}_0$ is $e^{{\alpha }_0}{s}_{{\alpha }_0}^{-1}=t_0{s}_{{\alpha }_0}^{-1}\text{.}$ A similar reasoning gives that the image of ${\bar{s}}_i\in A_{W_a}$ is $s_i\in A_{\stackrel{\sim }{W}}\text{.}$ $\square$

(5.8) Corollary. If $A_{\stackrel{\sim }{W}}$ is an extended Artin group of rank $\ell ,$ such that $W$ is finite and irreducible, then $A_{\stackrel{\sim }{W}}$ has a finite presentation with $\ell +1$ generators.

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

(5.9) Theorem. If $A_{\stackrel{\sim }{W}}$ is an extended Artin group of rank $\ell$ with $n_{ij}\in \{0,-1\}$ for all $i,j\in \{1,\dots ,\ell \}$ and with a connected Dynkin diagram, then $A_{\stackrel{\sim }{W}}$ has a finite presentation with $\ell +1$ generators.

		Proof.
	If $n_{ij}=-1$ then $s_i{t}_j{s}_i=t_i{t}_j$ (2.9), so then $t_i$ can be eliminated. Since the Dynkin diagram is connected all but one of the generators $t_1,\dots ,t_{\ell }$ can be eliminated. $\square$

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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