hr is minimal in the sense that one cannot delete any

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Unformatted text preview: reflections with respect to the lines − − forming the angle obtained from the angle ∠(H1 , H2 ), subdividing it into n equal parts. So, we may assume that φ = π/m. Case 2 : The angle φ is not of the form rπ for any rational r . In this case s2 s1 is of infinite order, G is isomorphic to D∞ , but it does not act discretely. Now suppose we have a convex polygon given as the intersection of a finite set of halfplanes r − Hi . P= i=1 We assume that the interior P o is not empty and the set {H1 , . . . , Hr } is minimal in the sense that one cannot delete any of the halfplanes without changing P . More importantly, we assume that − − ∠(Hi , Hj ) = π/mij for some positive integer mij or equal to 0 (mij = ∞). Let G be the group generated by reflections with mirror lines Hi . It is a discrete group of motions of the plane. The polygon P is a fundamental domain of G in the plane. Conversely any discrete group of motions of the plane generated by reflections is obtained in this way. Let p1 , . . . , pr be the vertices of the polygon P . We may assume that pi = Hi ∩ Hi+1 , where Hr+1 = H1 . Let mi = mii+1 . Since r (π/mi ) = (r − 2)π i=1 we have r i=1 1 = r − 2. mi The only solutions for (r ; m1 , . . . , mr ) are (3; 2, 3, 6), (3; 2, 4, 4), (3; 3, 3, 3), (4; 2, 2, 2, 2). 6 IGOR V. DOLGACHEV c • • • cc cc  cc  ccc  cc  cc    cc cc  ccc  ccc  c c  c  •c • • cc  cc  ccc  cc cc ''  c  c  ccc  ccc  cc  cc    • • • ∼ Z2 D8 (3; 2, 4, 4), G = •I RR  I R   III RR  RR I  II  •I •   II  RRR  II  II R   I I  RR   • RR  II   III  RR II   I  •RR • III   RR  I   R I  • ∼ (3; 3, 3, 3), G = Z2 S3 •I GG rrvvv RRR I R rrGrr II vvvvvR GG   I v R v rr I Iv •rvv GG II rrRR •    II v v GG  rrI r RR v v r Ir   I  G   R  vvr II G I   RR    • rrv RR  Irrrr GGGvvv II  II   RRrrr I  G vvvII v G v TT  r •Rvvv II • r RT v  I GG rrr TR v GG  vvII  rrr  RR TT  v  r  •r  (3; 2, 3, 6), G ∼ Z2 D12 = • • • • • 00 00 00 000 000 000 • • • • ∼ Z2 D4 (4; 2, 2, 2, 2), G = Figure 5. 2.2. Spaces of constant curvature. The usual euclidean plane is an example of a 2-dimensional space of zero constant curvature. Recall that a space of constant curvature is a simply connected Riemannian homogeneous space X such that the isotropy subgroup of its group of isometries Iso(X ) at each point coincides with the full orthogonal group of the tangent space. Up to isometry and rescaling the metric, there are three spaces of constant curvature of fixed dimension n. • The euclidean space E n with Iso(X ) equal to the affine orthogonal group AOn = Rn O(n). • The n-dimensional sphere S n = {(x0 , . . . , xn ) ∈ Rn+1 : x2 + . . . + x2 = 1} 0 n with Iso(X ) equal to the orthogonal group O(n + 1). • The hyperbolic (or Lobachevsky ) space H...
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