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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:
   
Document Type:
Article
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
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http://mi.mathnet.ru/eng/izv1742 http://mi.mathnet.ru/eng/izv/v44/i3/p637
Citing articles on Google Scholar:
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English citations
Related articles on Google Scholar:
Russian articles,
English articles
This publication is cited in the following articles:
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V. V. Nikulin, “On the classification of arithmetic groups generated by reflections in Lobachevsky spaces”, Math. USSR-Izv., 18:1 (1982), 99–123
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A. Neumaier, J. J. Seidel, “Discrete hyperbolic geometry”, Combinatorica, 3:2 (1983), 219
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È. B. Vinberg, “Hyperbolic reflection groups”, Russian Math. Surveys, 40:1 (1985), 31–75
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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”, Math. USSR-Sb., 66:1 (1990), 231–248
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C. Maclachlan, A. W. Reid, “The arithmetic structure of tetrahedral groups of hyperbolic isometries”, Mathematika, 36:2 (1989), 221
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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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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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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. 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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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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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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V.V.. Nikulin, “Elliptic Fibrations On K3 Surfaces”, Proceedings of the Edinburgh Mathematical Society, 2013, 1
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Belolipetsky M., “Arithmetic hyperbolic reflection groups”, Bull. Amer. Math. Soc., 53:3 (2016), 437–475
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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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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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