In this case the set of bounding hyperplanes can be

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Unformatted text preview: the angle φ = ∠(H1 , H2 ) = ∠(H1 , H2 ) is defined by cos φ = −(e1 , e2 ), 0 ≤ φ ≤ π. If X n = H n we use the same definition if (e1 , e2 ) ≤ 1; otherwise we say that the angle is divergent (in this case (e1 , e2 ) is equal to the hyperbolic cosine of the distance between the hyperplanes). 8 IGOR V. DOLGACHEV EE " EE " E" ""EE "" EE "" EE " EE "" EE " EE ~z{ xy}| "" " EE " EE "" E "" divergent GG GG  GG  GG  GG   ~Gz{ xy}| G  GGG G GG  GG   G   angle Figure 6. A convex polytope in X n is a nonempty intersection of a locally finite1 set of halfspaces − P = ∩i∈I Hi . − The normal vectors ei defining the halfspaces Hi all look inside the polytope. The hyperplanes Hi ’s are called faces of the polytope. In the case X n = H n we also add to P the points of the intersection lying on the absolute. We will always assume that no ei is a positive linear combination of others or, equivalently, none of the halfspaces contains the intersection of the rest of the halfspaces. In this case the set of bounding hyperplanes can be reconstructed from P . A convex polytope has a finite volume if and only if it is equal to a convex hull of finitely many points (vertices) from X n (or from the absolute if X n = H n ). Such a polytope has finitely many faces. If X n = E n or S n , it is a compact polytope. A polytope of finite volume in H n is compact only if its vertices do not lie on the absolute. 2.3. Reflection groups. A reflection group in a space of constant curvature is a discrete group of motions of X n generated by reflections. Theorem 2.1. Let Γ be a reflection group in X n . There exists a convex polytope − P (Γ) = ∩i∈I Hi such that (i) P is a fundamental domain for the action of Γ in X n ; − − (ii) the angle between any two halfspaces Hi , Hj is equal to zero or π/mij for some positive integer mij unless the angle is divergent; (iii) Γ is generated by reflections rHi , i ∈ I . Conversely, for every convex polytope P satisfying property (ii) the group Γ(P ) generated by the reflections into its facets is a reflection group and P satisfies (i). Proof. Consider the set H of mirror hyperplanes of all reflections contained in Γ. For any mirror hyperplane H and g ∈ Γ, the hyperplane g (H ) is the mirror hyperplane for the reflection grH g −1 . Thus the set H is invariant with respect to Γ. Let K be a compact subset of X n . For any hyperplane H ∈ H meeting K , we have rH (K ) ∩ K = ∅. Since Γ is a discrete group, the set {g ∈ G : g (K ) ∩ K = ∅} 1 Locally finite means that each compact subset of X n is intersected by only finitely many hyperplanes. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 9 is finite. This shows that the set H is locally finite. The closure of a connected component of H Xn \ H ∈H is a convex polytope called a cell (or Γ-cell or a fundamental polyhedron...
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