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The \(K(\pi, 1)\)-problem for hyperplane complements associated to infinite reflection groups. (English) Zbl 0833.51006
The authors generalize some results for finite Coxeter groups to infinite Coxeter groups. First, they introduce the notion of an infinite linear reflection group. A linear reflection on a real vector space \(V\) is a linear involution whose fixed subspace is a hyperplane. Let \(\overline {C}\) be a polyhedral cone of full dimension in \(V\) with interior \(C\), and let \(H_1, \dots, H_k\) be the hyperplanes spanned by the codimension one faces of \(C\). Denote by \(s_i\) a linear reflection with fixed hyperplane \(H_i\), and let \(W\) be the subgroup of \(\text{GL} (V)\) generated by \(\{s_1, \dots, s_k\}\). Then \(W\) is called a linear reflection group if \(w(C) \cap C = \emptyset\) for all \(w \in W\) with \(w \neq 1\). A theorem of Vinberg’s states that \((W, \{s_i\})\) is a Coxeter system, and that \(\overline {I} = \bigcup_{w\in W} w\overline {C}\) is a convex cone. The domain \(\Omega\) in \(V \otimes \mathbb{C}\) is defined by \(\Omega = \{v \in V \otimes \mathbb{C}\mid \text{Im}(v) \in I\} = V + iI\). By Vinberg’s theorem, \(W\) acts properly on \(\Omega\) and freely on (\(\Omega\)- \(\cup\) reflection hyperplanes). Define the manifold \(M\) by \(M = (\Omega\)- \(\cup\) reflection hyperplanes)\(/W\). The authors conjecture that \(M\) is an Eilenberg-MacLane space (as Deligne proved in the analogous situation for finite reflection groups). To prove this conjecture, the authors reformulate it in various different ways. As a first step, they show that it depends only on the Coxeter system \((W,S)\) whether this conjecture is true or not. Associated to the Coxeter system \((W,S)\) there is an Artin group \(A\) to which the authors define a certain simplicial complex \(\Phi\), called the modified Deligne complex. The authors prove that \(\Phi\) is simply connected and that the above mentioned conjecture is equivalent to the contractibility of \(\Phi\). The problem of the contractibility of \(\Phi\) is tackled by giving \(\Phi\) a piecewise Eulerian structure. Finally, the following theorems are shown: The manifold \(M\) is an Eilenberg-MacLane space for all Coxeter groups whose Coxeter system \((W,S)\) satisfies the condition
(FC) Assume \(T \subset S\), and every pair of elements in \(T\) generates a finite subgroup of \(W\). Then \(T\) generates a finite subgroup of \(W\). Moreover, \(M\) is an Eilenberg-MacLane space for all Coxeter systems such that the fundamental chamber \(K\) is two-dimensional.

MSC:
51F15 Reflection groups, reflection geometries
20F36 Braid groups; Artin groups
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