Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

 
 

 

Classification of $ 2$-reflective hyperbolic lattices of rank $ 4$


Author: 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
DOI: https://doi.org/10.1090/S0077-1554-07-00160-4
Published electronically: October 29, 2007
MathSciNet review: 2429266
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A hyperbolic lattice is said to be $ 2$-reflective if its automorphism group contains a subgroup of finite index generated by $ 2$-reflections. We determine all $ 2$-reflective hyperbolic lattices of rank $ 4$. (For all other values of the rank, this was done by V. V. Nikulin.)


References [Enhancements On Off] (What's this?)

  • 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 $ 2$-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. -, $ K3$ 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 $ 2$-reflective hyperbolic lattices of rank $ 4$. 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: https://doi.org/10.1090/S0077-1554-07-00160-4
Published electronically: October 29, 2007
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society