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Izv. Akad. Nauk SSSR Ser. Mat., 1980, Volume 44, Issue 3, Pages 637–669 (Mi izv1742)  

This article is cited in 27 scientific papers (total in 27 papers)

On arithmetic groups generated by reflections in Lobachevskii spaces

V. V. Nikulin

Abstract: Using È.  B. Vinberg's arithmeticity criterion, the author defines the notion of the Galois lattice of a discrete arithmetic group generated by reflections in a Lobachevskii space.
The author proves finiteness of the set of such lattices and, as a corollary, finiteness of the set of maximal discrete arithmetic groups generated by reflections for fixed dimension of the Lobachevskii space and fixed degree of the ground field over $\mathbf Q$.
Bibliography: 19 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1981, 16:3, 573–601

Bibliographic databases:

UDC: 519.46 + 511.4
MSC: Primary 20F32, 51F15; Secondary 52A25
Received: 12.11.1979
Revised: 11.12.1979

Citation: V. V. Nikulin, “On arithmetic groups generated by reflections in Lobachevskii spaces”, Izv. Akad. Nauk SSSR Ser. Mat., 44:3 (1980), 637–669; Math. USSR-Izv., 16:3 (1981), 573–601

Citation in format AMSBIB
\by V.~V.~Nikulin
\paper On arithmetic groups generated by reflections in Lobachevskii spaces
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1980
\vol 44
\issue 3
\pages 637--669
\jour Math. USSR-Izv.
\yr 1981
\vol 16
\issue 3
\pages 573--601

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    This publication is cited in the following articles:
    1. V. V. Nikulin, “On the classification of arithmetic groups generated by reflections in Lobachevsky spaces”, Math. USSR-Izv., 18:1 (1982), 99–123  mathnet  crossref  mathscinet  zmath  isi
    2. A. Neumaier, J. J. Seidel, “Discrete hyperbolic geometry”, Combinatorica, 3:2 (1983), 219  crossref  mathscinet  zmath  isi
    3. È. B. Vinberg, “Hyperbolic reflection groups”, Russian Math. Surveys, 40:1 (1985), 31–75  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. A. G. Khovanskii, “Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups in Lobachevskii space”, Funct. Anal. Appl., 20:1 (1986), 41–50  mathnet  crossref  mathscinet  zmath  isi
    5. V. V. Nikulin, “Del Pezzo surfaces with log-terminal singularities”, Math. USSR-Sb., 66:1 (1990), 231–248  mathnet  crossref  mathscinet  zmath  isi
    6. C. Maclachlan, A. W. Reid, “The arithmetic structure of tetrahedral groups of hyperbolic isometries”, Mathematika, 36:2 (1989), 221  crossref  isi
    7. V. V. Nikulin, “Reflection groups in Lobachevskii spaces and the denominator identity for Lorentzian Kac–Moody algebras”, Izv. Math., 60:2 (1996), 305–334  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. V. A. Gritsenko, V. V. Nikulin, “Igusa modular forms and 'the simplest' Lorentzian Kac–Moody algebras”, Sb. Math., 187:11 (1996), 1601–1641  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. Gritsenko V.A., Nikulin V.V., “Automorphic forms and Lorentzian Kac-Moody algebras. Part I”, International Journal of Mathematics, 9:2 (1998), 153–199  crossref  isi  elib
    10. V. V. Nikulin, “On the Classification of Hyperbolic Root Systems of Rank Three”, Proc. Steklov Inst. Math., 230:3 (2000), 1–241  mathnet  mathscinet  zmath
    11. Gritsenko V.A., Nikulin V.V., “The arithmetic mirror symmetry and Calabi-Yau manifolds”, Communications in Mathematical Physics, 210:1 (2000), 1–11  crossref  isi  elib
    12. V. A. Gritsenko, V. V. Nikulin, “On classification of Lorentzian Kac–Moody algebras”, Russian Math. Surveys, 57:5 (2002), 921–979  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. V. V. Nikulin, “Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces”, Izv. Math., 71:1 (2007), 53–56  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. V. V. Nikulin, “On Ground Fields of Arithmetic Hyperbolic Reflection Groups. II”, Mosc. Math. J., 8:4 (2008), 789–812  mathnet  crossref  mathscinet  zmath
    15. Agol, I, “Finiteness of arithmetic hyperbolic reflection groups”, Groups Geometry and Dynamics, 2:4 (2008), 481  isi
    16. Nikulin, VV, “On ground fields of arithmetic hyperbolic reflection groups. III”, Journal of the London Mathematical Society-Second Series, 79 (2009), 738  isi
    17. Nikulin V.V., “On Ground Fields of Arithmetic Hyperbolic Reflection Groups”, Groups and Symmetries: From Neolithic Scots To John McKay, CRM Proceedings & Lecture Notes, 47, 2009, 299–326  isi
    18. Maclachlan C., “Bounds for discrete hyperbolic arithmetic reflection groups in dimension 2”, Bull London Math Soc, 43:1 (2011), 111–123  crossref  isi
    19. Viacheslav V. Nikulin, “Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups”, Proc. Steklov Inst. Math., 273 (2011), 229–237  mathnet  crossref  mathscinet  zmath  isi  elib
    20. V. V. Nikulin, “The transition constant for arithmetic hyperbolic reflection groups”, Izv. Math., 75:5 (2011), 971–1005  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    21. Maclachlan C., “Commensurability classes of discrete arithmetic hyperbolic groups”, Groups Geom Dyn, 5:4 (2011), 767–785  isi
    22. V.V.. Nikulin, “Elliptic Fibrations On K3 Surfaces”, Proceedings of the Edinburgh Mathematical Society, 2013, 1  crossref
    23. Belolipetsky M., “Arithmetic hyperbolic reflection groups”, Bull. Amer. Math. Soc., 53:3 (2016), 437–475  crossref  mathscinet  zmath  isi  elib  scopus
    24. Gritsenko V. Nikulin V.V., “Lorentzian Kac-Moody Algebras With Weyl Groups of 2-Reflections”, Proc. London Math. Soc., 116:3 (2018), 485–533  crossref  isi
    25. Mark A., “The Classification of Rank 3 Reflective Hyperbolic Lattices Over Z[Root 2]”, Math. Proc. Camb. Philos. Soc., 164:2 (2018), 221–257  crossref  isi
    26. Turkalj I., “Reflective Lorentzian Lattices of Signature (5,1)”, J. Algebra, 513 (2018), 516–544  crossref  mathscinet  zmath  isi  scopus
    27. N. V. Bogachev, “Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank 4”, Izv. Math., 83:1 (2019), 1–19  mathnet  crossref  crossref  adsnasa  isi  elib
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