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Izv. Akad. Nauk SSSR Ser. Mat., 1971, Volume 35, Issue 5, Pages 1072–1112 (Mi izv2148)  

This article is cited in 36 scientific papers (total in 37 papers)

Discrete linear groups generated by reflections

È. B. Vinberg


Abstract: We investigate linear groups generated by reflections in the faces of a convex polyhedral cone and operating discretely on an open convex cone. These groups generalize the discrete groups of motions in simply connected spaces of constant curvature, generated by reflection. Like the latter, they turn out to be Coxeter groups.

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English version:
Mathematics of the USSR-Izvestiya, 1971, 5:5, 1083–1119

Bibliographic databases:

UDC: 519.45
MSC: Primary 20H15, 50C15; Secondary 52F05, 52A25, 50C20
Received: 16.06.1970

Citation: È. B. Vinberg, “Discrete linear groups generated by reflections”, Izv. Akad. Nauk SSSR Ser. Mat., 35:5 (1971), 1072–1112; Math. USSR-Izv., 5:5 (1971), 1083–1119

Citation in format AMSBIB
\Bibitem{Vin71}
\by \`E.~B.~Vinberg
\paper Discrete linear groups generated by reflections
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1971
\vol 35
\issue 5
\pages 1072--1112
\mathnet{http://mi.mathnet.ru/izv2148}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=302779}
\zmath{https://zbmath.org/?q=an:0247.20054}
\transl
\jour Math. USSR-Izv.
\yr 1971
\vol 5
\issue 5
\pages 1083--1119
\crossref{https://doi.org/10.1070/IM1971v005n05ABEH001203}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. È. B. Vinberg, “On unimodular integral quadratic forms”, Funct. Anal. Appl., 6:2 (1972), 105–111  mathnet  crossref  mathscinet  zmath
    2. V.G Kac, “Infinite-dimensional algebras, Dedekind's η-function, classical möbius function and the very strange formula”, Advances in Mathematics, 30:2 (1978), 85  crossref
    3. V. G. Kac, “Infinite root systems, representations of graphs and invariant theory”, Invent math, 56:1 (1980), 57  crossref  mathscinet  zmath  adsnasa  isi
    4. V. V. Nikulin, “On arithmetic groups generated by reflections in Lobachevskii spaces”, Math. USSR-Izv., 16:3 (1981), 573–601  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. George Maxwell, “Sphere packings and hyperbolic reflection groups”, Journal of Algebra, 79:1 (1982), 78  crossref
    6. Seppo Rickman, “Quasiregular mappings and metrics on then-sphere with punctures”, Comment Math Helv, 59:1 (1984), 136  crossref  mathscinet  zmath  isi
    7. Victor G Kač, Dale H Peterson, “Infinite-dimensional Lie algebras, theta functions and modular forms”, Advances in Mathematics, 53:2 (1984), 125  crossref
    8. È. B. Vinberg, “Hyperbolic reflection groups”, Russian Math. Surveys, 40:1 (1985), 31–75  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. Boris Kimelfeld, “Homogeneous domains on flag manifolds”, Journal of Mathematical Analysis and Applications, 121:2 (1987), 506  crossref
    10. C Saclioglu, J Phys A Math Gen, 22:18 (1989), 3753  crossref  mathscinet  zmath  adsnasa
    11. Ruth Charney, “Artin groups of finite type are biautomatic”, Math Ann, 292:1 (1992), 671  crossref  mathscinet  zmath  isi
    12. Giovanni Zanzotto, “On the material symmetry group of elastic crystals and the Born Rule”, Arch Rational Mech Anal, 121:1 (1992), 1  crossref  mathscinet  zmath
    13. Roger C. Alperin, “Free products of matrices”, Linear and Multilinear Algebra, 38:3 (1995), 215  crossref
    14. R. B. Howlett, P. J. Rowley, D. E. Taylor, “On outer automorphism groups of Coxeter groups”, manuscripta math, 93:1 (1997), 499  crossref  mathscinet  zmath  isi
    15. D. V. Alekseevskii, V. O. Bugaenko, G. I. Olshanskii, V. L. Popov, O. V. Schwarzman, “Érnest Borisovich Vinberg (on his 60th birthday)”, Russian Math. Surveys, 52:6 (1997), 1335–1343  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    16. M Salvetti, “Artin groups associated to infinite Coxeter groups”, Discrete Mathematics, 163:1-3 (1997), 129  crossref  elib
    17. C. De Concini, M. Salvetti, F. Stumbo, “The top-cohomology of Artin groups with coefficients in rank-1 local systems over”, Topology and its Applications, 78:1-2 (1997), 5  crossref
    18. I. I. Kotova, “Description of root systems with given Weyl group”, Math. Notes, 64:3 (1998), 342–346  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    19. A.N. Dranishnikov, “Boundaries of Coxeter groups and simplicial complexes with given links”, Journal of Pure and Applied Algebra, 137:2 (1999), 139  crossref
    20. Mitsuko Abe, Masamichi Sato, “Calabi-Yau manifolds by Borcea-Voisin method”, J Math Phys (N Y ), 41:5 (2000), 3045  crossref  mathscinet  zmath
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    22. P. V. Tumarkin, “Maximum-rank root subsystems of hyperbolic root systems”, Sb. Math., 195:1 (2004), 121–134  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    23. R. M. Garipov, “Crystallographic classes in a 4-dimensional Minkovskii space”, Algebra and Logic, 47:1 (2008), 18–31  mathnet  crossref  mathscinet  zmath  isi  elib
    24. A. A. Gaifullin, “The Manifold of Isospectral Symmetric Tridiagonal Matrices and Realization of Cycles by Aspherical Manifolds”, Proc. Steklov Inst. Math., 263 (2008), 38–56  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    25. Ludovic Marquis, “Espace des modules de certains polyèdres projectifs miroirs”, Geom Dedicata, 2009  crossref
    26. Allcock D., Carlson J.A., Toledo D., “Hyperbolic Geometry and Moduli of Real Cubic Surfaces”, Annales Scientifiques de l Ecole Normale Superieure, 43:1 (2010), 69–115  isi
    27. Robert Donnelly, “Root Systems for Asymmetric Geometric Representations of Coxeter Groups”, Comm. in Algebra, 39:4 (2011), 1298  crossref
    28. Anna Felikson, Pavel Tumarkin, “Hyperbolic subalgebras of hyperbolic Kac–Moody algebras”, Transformation Groups, 2011  crossref
    29. T. E. Panov, “Geometric structures on moment-angle manifolds”, Russian Math. Surveys, 68:3 (2013), 503–568  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    30. Daniel Allcock, “Reflection centralizers in Coxeter groups”, Transformation Groups, 2013  crossref
    31. Trans. Moscow Math. Soc., 74 (2013), 203–216  mathnet  crossref  mathscinet  zmath  elib
    32. Gaifullin A., “Universal Realisators for Homology Classes”, Geom. Topol., 17:3 (2013), 1745–1772  crossref  isi
    33. Kanghyun Choi, Suhyoung Choi, “The definability criteria for convex projective polyhedral reflection groups”, Geom Dedicata, 2014  crossref
    34. N. Yu. Erokhovets, “Buchstaber invariant theory of simplicial complexes and convex polytopes”, Proc. Steklov Inst. Math., 286 (2014), 128–187  mathnet  crossref  crossref  isi  elib  elib
    35. E. B. Vinberg, M. A. Jibladze, A. G. Elashvili, “Moduli algebras of some non-semiquasihomogeneous singularities”, Funct. Anal. Appl., 51:2 (2017), 86–97  mathnet  crossref  crossref  isi  elib
    36. Ishida M., “Cusp Singularities and Quasi-Polyhedral Sets”, Algebraic Varieties and Automorphism Groups, Advanced Studies in Pure Mathematics, 75, ed. Masuda K. Kishimoto T. Kojima H. Miyanishi M. Zaidenberg M., Math Soc Japan, 2017, 163–182  isi
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  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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