Classification of 2-reflective hyperbolic lattices of rank 4

Vinberg, E.

doi:10.1090/S0077-1554-07-00160-4

Classification of $2$-reflective hyperbolic lattices of rank $4$
HTML articles powered by AMS MathViewer

by E. B. Vinberg
Translated by: Alex Martsinkovsky

Trans. Moscow Math. Soc. 2007, 39-66

DOI: https://doi.org/10.1090/S0077-1554-07-00160-4

Published electronically: October 29, 2007

PDF | Request permission

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

E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
V. O. Bugaenko, Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices, Lie groups, their discrete subgroups, and invariant theory, Adv. Soviet Math., vol. 8, Amer. Math. Soc., Providence, RI, 1992, pp. 33–55. MR 1155663, DOI 10.1007/bf00883263
J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447, DOI 10.1007/978-1-4757-6568-7
V. V. Nikulin, Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by $2$-reflections. Algebro-geometric applications, Current problems in mathematics, Vol. 18, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1981, pp. 3–114 (Russian). MR 633160
V. V. Nikulin, $K3$ surfaces with a finite group of automorphisms and a Picard group of rank three, Trudy Mat. Inst. Steklov. 165 (1984), 119–142 (Russian). Algebraic geometry and its applications. MR 752938
V. V. Nikulin, Discrete reflection groups in Lobachevsky spaces and algebraic surfaces, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 654–671. MR 934268
Rudolf Scharlau and Claudia Walhorn, Integral lattices and hyperbolic reflection groups, Astérisque 209 (1992), 15–16, 279–291. Journées Arithmétiques, 1991 (Geneva). MR 1211022
È. B. Vinberg, Some examples of crystallographic groups in Lobačevskiĭ spaces, Mat. Sb. (N.S.) 78 (120) (1969), 633–639 (Russian). MR 0246193
È. B. Vinberg, The groups of units of certain quadratic forms, Mat. Sb. (N.S.) 87(129) (1972), 18–36 (Russian). MR 0295193
È. B. Vinberg, The unimodular integral quadratic forms, Funkcional. Anal. i Priložen. 6 (1972), no. 2, 24–31 (Russian). MR 0299557
È. B. Vinberg, Some arithmetical discrete groups in Lobačevskiĭ spaces, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 323–348. MR 0422505
È. B. Vinberg, Absence of crystallographic groups of reflections in Lobachevskiĭ spaces of large dimension, Trudy Moskov. Mat. Obshch. 47 (1984), 68–102, 246 (Russian). MR 774946
—, Classification of $2$-reflective hyperbolic lattices of rank $4$. Preprint 98– 113. Universität Bielefeld. 1998.