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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 • • ... • • ˜
A Bn , Cn • • ... • • ˜
r • ... • • •
• • . . . • vvv
• • • • ... • • • ˜
r . . . • vvv
• E6 •••••
E6 E7 ••••••
I2 (m) • m • m≥5 H3 ˜
E8 ••• H4 ˜
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 ). It is known that they do not exist
if n ≥ 996 (, , ) and even if n > 300 (see the announcement in ).
There are no cocompact reﬂection groups in H n for n ≥ 30 .
One can give the following description of Coxeter diagrams deﬁning reﬂection
groups of coﬁnite volume (see ).
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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