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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
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\jour Math. USSR-Izv.
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\pages 99--123
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http://mi.mathnet.ru/eng/izv1550 http://mi.mathnet.ru/eng/izv/v45/i1/p113
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This publication is cited in the following articles:
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È. B. Vinberg, “Absence of crystallographic groups of reflections in Lobachevskii spaces of large dimension”, Funct. Anal. Appl., 15:2 (1981), 128–130
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È. B. Vinberg, “Hyperbolic reflection groups”, Russian Math. Surveys, 40:1 (1985), 31–75
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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
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A. G. Khovanskii, “Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups in Lobachevskii space”, Funct. Anal. Appl., 20:1 (1986), 41–50
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V. V. Nikulin, “Del Pezzo surfaces with log-terminal singularities. II”, Math. USSR-Izv., 33:2 (1989), 355–372
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V. V. Nikulin, “Del Pezzo surfaces with log-terminal singularities. III”, Math. USSR-Izv., 35:3 (1990), 657–675
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V. V. Nikulin, “Del Pezzo surfaces with log-terminal singularities”, Math. USSR-Sb., 66:1 (1990), 231–248
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V. V. Nikulin, “Algebraic three-folds and the diagram method”, Math. USSR-Izv., 37:1 (1991), 157–189
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V. A. Gritsenko, V. V. Nikulin, “Igusa modular forms and 'the simplest' Lorentzian Kac–Moody algebras”, Sb. Math., 187:11 (1996), 1601–1641
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V. V. Nikulin, “Reflection groups in Lobachevskii spaces and the denominator identity for Lorentzian Kac–Moody algebras”, Izv. Math., 60:2 (1996), 305–334
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Gritsenko V.A., Nikulin V.V., “Automorphic forms and Lorentzian Kac-Moody algebras. Part I”, International Journal of Mathematics, 9:2 (1998), 153–199
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V. V. Nikulin, “On the Classification of Hyperbolic Root Systems of Rank Three”, Proc. Steklov Inst. Math., 230:3 (2000), 1–241
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Gritsenko V.A., Nikulin V.V., “The arithmetic mirror symmetry and Calabi-Yau manifolds”, Communications in Mathematical Physics, 210:1 (2000), 1–11
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V. A. Timorin, “On Polytopes that are Simple at the Edges”, Funct. Anal. Appl., 35:3 (2001), 189–198
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V. A. Gritsenko, V. V. Nikulin, “On classification of Lorentzian Kac–Moody algebras”, Russian Math. Surveys, 57:5 (2002), 921–979
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V. V. Nikulin, “Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces”, Izv. Math., 71:1 (2007), 53–56
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P. V. Tumarkin, A. A. Felikson, “On simple ideal hyperbolic Coxeter polytopes”, Izv. Math., 72:1 (2008), 113–126
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V. V. Nikulin, “On Ground Fields of Arithmetic Hyperbolic Reflection Groups. II”, Mosc. Math. J., 8:4 (2008), 789–812
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Agol, I, “Finiteness of arithmetic hyperbolic reflection groups”, Groups Geometry and Dynamics, 2:4 (2008), 481
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Nikulin, VV, “On ground fields of arithmetic hyperbolic reflection groups. III”, Journal of the London Mathematical Society-Second Series, 79 (2009), 738
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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
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Guillaume Dufour, “Notes on right-angled Coxeter polyhedra in hyperbolic spaces”, Geom Dedicata, 2010
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Maclachlan C., “Bounds for discrete hyperbolic arithmetic reflection groups in dimension 2”, Bull London Math Soc, 43:1 (2011), 111–123
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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
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V. V. Nikulin, “The transition constant for arithmetic hyperbolic reflection groups”, Izv. Math., 75:5 (2011), 971–1005
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Maclachlan C., “Commensurability classes of discrete arithmetic hyperbolic groups”, Groups Geom Dyn, 5:4 (2011), 767–785
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Mark Pollicott, Richard Sharp, “Correlations of Length Spectra for Negatively Curved Manifolds”, Commun. Math. Phys, 2012
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V.V.. Nikulin, “Elliptic Fibrations On K3 Surfaces”, Proceedings of the Edinburgh Mathematical Society, 2013, 1
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Belolipetsky M. Linowitz B., “On Fields of Definition of Arithmetic Kleinian Reflection Groups II”, Int. Math. Res. Notices, 2014, no. 9, 2559–2571
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Nonaka J., “The Number of Cusps of Right-angled Polyhedra in Hyperbolic Spaces”, Tokyo J. Math., 38:2 (2015), 539–560
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Belolipetsky M., “Arithmetic hyperbolic reflection groups”, Bull. Amer. Math. Soc., 53:3 (2016), 437–475
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N. V. Bogachev, “Reflective anisotropic hyperbolic lattices of rank 4”, Russian Math. Surveys, 72:1 (2017), 179–181
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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
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A. Yu. Vesnin, “Right-angled polyhedra and hyperbolic 3-manifolds”, Russian Math. Surveys, 72:2 (2017), 335–374
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Gritsenko V. Nikulin V.V., “Lorentzian Kac-Moody Algebras With Weyl Groups of 2-Reflections”, Proc. London Math. Soc., 116:3 (2018), 485–533
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Mark A., “The Classification of Rank 3 Reflective Hyperbolic Lattices Over Z[Root 2]”, Math. Proc. Camb. Philos. Soc., 164:2 (2018), 221–257
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Linowitz B., “Bounds For Arithmetic Hyperbolic Reflection Groups in Dimension 2”, Transform. Groups, 23:3 (2018), 743–753
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V. A. Gritsenko, “Reflective modular forms and applications”, Russian Math. Surveys, 73:5 (2018), 797–864
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Turkalj I., “Reflective Lorentzian Lattices of Signature (5,1)”, J. Algebra, 513 (2018), 516–544
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N. V. Bogachev, “Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank 4”, Izv. Math., 83:1 (2019), 1–19
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