In this case the set of bounding hyperplanes can be

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online