|
Classification of -reflective hyperbolic lattices of rank
Author(s):
E.
B.
Vinberg
Translated by:
Alex Martsinkovsky
Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva,
tom 68 (2007).
Journal:
Trans. Moscow Math. Soc.
2007,
39-66.
MSC (2000):
Primary 11H06
Posted:
October 29, 2007
MathSciNet review:
2429266
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
A hyperbolic lattice is said to be -reflective if its automorphism group contains a subgroup of finite index generated by -reflections. We determine all -reflective hyperbolic lattices of rank . (For all other values of the rank, this was done by V. V. Nikulin.)
References:
-
- 1.
- Andreev, E.M., Convex polyhedra of finite volume in Lobachevsky space. (Russian), Mat. Sb. (N.S.) 83(125) (1970), 256-260. MR 0273510 (42:8388)
- 2.
- Bugaenko, V.O., Arithmetic crystallographic groups generated by reflections and reflective hyperbolic lattices. Adv. Sov. Math. 1992. Vol.8, E.B. Vinberg, ed., pp.33-55. MR 1155663 (93g:20094)
- 3.
- Cassels, J.W.S., Rational quadratic forms. London Math. Soc. Monographs, 13. Academic Press, London, New York, 1978. MR 522835 (80m:10019)
- 4.
- Conway, J. H., Sloane, N. J. A., Sphere packings, lattices and groups. Third edition. Springer-Verlag, New York, 1999. MR 1662447 (2000b:11077)
- 5.
- Nikulin, V,V., Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by
-reflections. Algebro-geometric applications. (Russian) Current problems in mathematics, Vol. 18, pp. 3-114, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981 MR 633160 (83c:10030) - 6.
- -,
surfaces with a finite group of automorphisms and a Picard group of rank three. (Russian) Algebraic geometry and its applications. Trudy Mat. Inst. Steklov. 165 (1984), 119-142. MR 0752938 (86e:14018) - 7.
- -, Discrete reflection groups in Lobachevsky spaces and algebraic surfaces. Proc. Int. Congr. Math. Berkeley. 1986. Vol.1., pp.654-671. MR 0934268 (89d:11032)
- 8.
- Scharlau R., Walhorn C., Integral lattices and hyperbolic reflection groups. Astérisque. 1992. Vol.209, pp.279-291. MR 1211022 (94j:11057)
- 9.
- Vinberg, E.B., Some examples of crystallographic groups in Lobachevsky spaces. (Russian) Mat. Sb. (N.S.) 78 (120) (1969), 633-639. MR 0246193 (39:7497)
- 10.
- -, The groups of units of certain quadratic forms. (Russian) Mat. Sb. (N.S.) 87(129) (1972), 18-36. MR 0295193 (45:4261)
- 11.
- -, The unimodular integral quadratic forms. (Russian) Funkcional. Anal. i Prilozen. 6 (1972), no. 2, 24-31. MR 0299557 (45:8605)
- 12.
- -, Some arithmetical discrete groups in Lobachevsky spaces. Proc. Int. Coll. on Discrete Subgroups of Lie Groups and Appl. to Moduli. Pap. Bombay Coll. 1975. pp.323-348. MR 0422505 (54:10492)
- 13.
- -, Absence of crystallographic groups of reflections in Lobachevsky spaces of large dimension. (Russian) Trudy Moskov. Mat. Obshch. 47 (1984), 68-102, 246. MR 774946 (86i:22020)
- 14.
- -, Classification of
-reflective hyperbolic lattices of rank . Preprint 98-113. Universität Bielefeld. 1998.
Similar Articles:
Retrieve articles in Transactions of the Moscow Mathematical Society
with MSC
(2000):
11H06
Retrieve articles in all Journals with MSC
(2000):
11H06
Additional Information:
E.
B.
Vinberg
Affiliation:
Department of Mechanics and Mathematics, Moscow State University, Moscow 119992, GSP-2, Russia
Email:
vinberg@ebv.pvt.msu.su
DOI:
10.1090/S0077-1554-07-00160-4
PII:
S 0077-1554(07)00160-4
Posted:
October 29, 2007
Copyright of article:
Copyright
2007,
American Mathematical Society
|