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Izv. Akad. Nauk SSSR Ser. Mat., 1981, Volume 45, Issue 1, Pages 113–142 (Mi izv1550)  

This article is cited in 40 scientific papers (total in 40 papers)

On the classification of arithmetic groups generated by reflections in Lobachevsky spaces

V. V. Nikulin

Abstract: It is proved that there do not exist discrete arithmetic groups generated by reflections in Lobachevsky spaces if the dimension of the Lobachevsky space is greater than 15 and the degree of the ground field is sufficiently large.
Bibliography: 24 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1982, 18:1, 99–123

Bibliographic databases:

UDC: 519.46+511.4
MSC: Primary 51F15, 20H15; Secondary 20F32, 51M10, 51M20, 52A25
Received: 08.07.1980

Citation: V. V. Nikulin, “On the classification of arithmetic groups generated by reflections in Lobachevsky spaces”, Izv. Akad. Nauk SSSR Ser. Mat., 45:1 (1981), 113–142; Math. USSR-Izv., 18:1 (1982), 99–123

Citation in format AMSBIB
\by V.~V.~Nikulin
\paper On the classification of arithmetic groups generated by reflections in Lobachevsky spaces
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1981
\vol 45
\issue 1
\pages 113--142
\jour Math. USSR-Izv.
\yr 1982
\vol 18
\issue 1
\pages 99--123

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    This publication is cited in the following articles:
    1. È. B. Vinberg, “Absence of crystallographic groups of reflections in Lobachevskii spaces of large dimension”, Funct. Anal. Appl., 15:2 (1981), 128–130  mathnet  crossref  mathscinet  zmath  isi
    2. È. B. Vinberg, “Hyperbolic reflection groups”, Russian Math. Surveys, 40:1 (1985), 31–75  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. M. N. Prokhorov, “The absence of discrete reflection groups with noncompact fundamental polyhedron of finite volume in Lobachevskii space of large dimension”, Math. USSR-Izv., 28:2 (1987), 401–411  mathnet  crossref  mathscinet  zmath
    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. II”, Math. USSR-Izv., 33:2 (1989), 355–372  mathnet  crossref  mathscinet  zmath
    6. V. V. Nikulin, “Del Pezzo surfaces with log-terminal singularities. III”, Math. USSR-Izv., 35:3 (1990), 657–675  mathnet  crossref  mathscinet  zmath
    7. V. V. Nikulin, “Del Pezzo surfaces with log-terminal singularities”, Math. USSR-Sb., 66:1 (1990), 231–248  mathnet  crossref  mathscinet  zmath  isi
    8. V. V. Nikulin, “Algebraic three-folds and the diagram method”, Math. USSR-Izv., 37:1 (1991), 157–189  mathnet  crossref  mathscinet  zmath  adsnasa
    9. 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
    10. 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
    11. 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
    12. 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
    13. 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
    14. V. A. Timorin, “On Polytopes that are Simple at the Edges”, Funct. Anal. Appl., 35:3 (2001), 189–198  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    15. 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
    16. 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
    17. P. V. Tumarkin, A. A. Felikson, “On simple ideal hyperbolic Coxeter polytopes”, Izv. Math., 72:1 (2008), 113–126  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    18. V. V. Nikulin, “On Ground Fields of Arithmetic Hyperbolic Reflection Groups. II”, Mosc. Math. J., 8:4 (2008), 789–812  mathnet  crossref  mathscinet  zmath
    19. Agol, I, “Finiteness of arithmetic hyperbolic reflection groups”, Groups Geometry and Dynamics, 2:4 (2008), 481  isi
    20. Nikulin, VV, “On ground fields of arithmetic hyperbolic reflection groups. III”, Journal of the London Mathematical Society-Second Series, 79 (2009), 738  isi
    21. 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
    22. Guillaume Dufour, “Notes on right-angled Coxeter polyhedra in hyperbolic spaces”, Geom Dedicata, 2010  crossref
    23. Maclachlan C., “Bounds for discrete hyperbolic arithmetic reflection groups in dimension 2”, Bull London Math Soc, 43:1 (2011), 111–123  crossref  isi
    24. 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
    25. 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
    26. Maclachlan C., “Commensurability classes of discrete arithmetic hyperbolic groups”, Groups Geom Dyn, 5:4 (2011), 767–785  isi
    27. Mark Pollicott, Richard Sharp, “Correlations of Length Spectra for Negatively Curved Manifolds”, Commun. Math. Phys, 2012  crossref
    28. V.V.. Nikulin, “Elliptic Fibrations On K3 Surfaces”, Proceedings of the Edinburgh Mathematical Society, 2013, 1  crossref
    29. Belolipetsky M. Linowitz B., “On Fields of Definition of Arithmetic Kleinian Reflection Groups II”, Int. Math. Res. Notices, 2014, no. 9, 2559–2571  crossref  isi
    30. Nonaka J., “The Number of Cusps of Right-angled Polyhedra in Hyperbolic Spaces”, Tokyo J. Math., 38:2 (2015), 539–560  crossref  mathscinet  zmath  isi  elib
    31. Belolipetsky M., “Arithmetic hyperbolic reflection groups”, Bull. Amer. Math. Soc., 53:3 (2016), 437–475  crossref  mathscinet  zmath  isi  elib  scopus
    32. N. V. Bogachev, “Reflective anisotropic hyperbolic lattices of rank 4”, Russian Math. Surveys, 72:1 (2017), 179–181  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    33. V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, S. Park, “Cohomological rigidity of manifolds defined by 3-dimensional polytopes”, Russian Math. Surveys, 72:2 (2017), 199–256  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    34. A. Yu. Vesnin, “Right-angled polyhedra and hyperbolic 3-manifolds”, Russian Math. Surveys, 72:2 (2017), 335–374  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    35. 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
    36. 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
    37. Linowitz B., “Bounds For Arithmetic Hyperbolic Reflection Groups in Dimension 2”, Transform. Groups, 23:3 (2018), 743–753  crossref  isi
    38. V. A. Gritsenko, “Reflective modular forms and applications”, Russian Math. Surveys, 73:5 (2018), 797–864  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    39. Turkalj I., “Reflective Lorentzian Lattices of Signature (5,1)”, J. Algebra, 513 (2018), 516–544  crossref  mathscinet  zmath  isi  scopus
    40. N. V. Bogachev, “Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank 4”, Izv. Math., 83:1 (2019), 1–19  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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