# Proposition 24 a reection group in h n is of nite

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Unformatted text preview: to the semi-direct product 2n Σn . It acts in the euclidean space Rn as a group generated by reﬂections in vectors ei − ei+1 , i = 1, . . . , n − 1, and en . The group of type Dn is isomorphic to the semi-direct product 2n−1 Σn . It acts in the euclidean space Rn as a group generated by reﬂections in vectors ei − ei+1 , i = 1, . . . , n − 1, and en−1 + en . REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 11 Table 1. Spherical and euclidean real reﬂection groups An • • ... • • ˜ A1 ˜n A Bn , Cn • • ... • • ˜ Bn ∞ •• • • .. .iiiii •• i • v • vvv • r • ... • • • rr •r ˜ Cn Dn • r r vrr • • . . . • vvv • • • • ... • • • ˜ Dn v • vvv • r r vrr • r . . . • vvv rrr • • E6 ••••• • ˜ E6 E7 •••••• • ••••••• • •••• ˜ E7 E8 F4 I2 (m) • m • m≥5 H3 ˜ G2 6 ••• ˜ E8 ••• H4 ˜ F4 ••••• • • ••••••• • •••••••• • ••••• •••• Spherical groups Euclidean groups A discrete group Γ of motions in X n admitting a fundamental domain of ﬁnite volume (resp. compact) is said to be of ﬁnite covolume (resp. cocompact ).2 Obviously, a simplex in S n or E n is compact. Thus the previous list gives a classiﬁcation of irreducible reﬂection groups of ﬁnite covolume in E n and S n . They are are automatically cocompact. The classiﬁcation of reﬂection groups of ﬁnite covolume in H n is known only for n = 2 (H. Poincar´) and n = 3 (E. Andreev [5]). It is known that they do not exist e if n ≥ 996 ([65], [66], [95]) and even if n &gt; 300 (see the announcement in [87]). There are no cocompact reﬂection groups in H n for n ≥ 30 [116]. One can give the following description of Coxeter diagrams deﬁning reﬂection groups of coﬁnite volume (see [112]). Proposition 2.4. A reﬂection group Γ in H n is of ﬁnite covolume if and only if any elliptic subdiagram of rank n − 1 of its Coxeter diagram can be extended in 2 If Γ is realized as a discrete subgroup of a Lie group that acts properly and transitively on X n , then this terminology agrees with the terminology of discrete subgroups of a Lie group. 12 IGOR V. DOLGACHEV exactly two ways to an elliptic subdiagram of rank n or a parabolic subdiagram of rank n − 1. Moreover, Γ is cocompact if the same is true but there are no parabolic subdiagrams of rank n − 1. The geometric content of this proposition is as follows. The intersection of hyperplanes deﬁning an elliptic subdiagram of rank n − 1 is of dimension 1 (a onedimensional facet of the polytope). An elliptic subdiagram (resp. parabolic) of rank n deﬁnes a proper (resp. improper) vertex of the polytope. So, the proposition says that Γ is of ﬁnite covolume if and only if each one-dimensional facet joins precisely two vertices, proper or improper. 2.4. Coxeter groups. Recall that a Coxeter group is a group W adm...
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## This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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