This preview shows page 1. Sign up to view the full content.
Unformatted text preview: to the semidirect 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 semidirect 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 > 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 onedimensional facet joins precisely
two vertices, proper or improper.
2.4. Coxeter groups. Recall that a Coxeter group is a group W adm...
View
Full
Document
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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

Click to edit the document details