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Izv. RAN. Ser. Mat., 2007, Volume 71, Issue 1, Pages 55–60 (Mi izv1148)  

This article is cited in 19 scientific papers (total in 19 papers)

Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces

V. V. Nikulinab

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Liverpool

Abstract: After results of the author (1980, 1981) and Vinberg (1981), the finiteness of the number of maximal arithmetic groups generated by reflections in Lobachevsky spaces remained unknown in dimensions $2\le n\le 9$ only. It was proved recently (2005) in dimension 2 by Long, Maclachlan and Reid and in dimension 3 by Agol. Here we use the results in dimensions 2 and 3 to prove the finiteness in all remaining dimensions $4\le n\le 9$. The methods of the author (1980, 1981) are more than sufficient for this using a very short and very simple argument.


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English version:
Izvestiya: Mathematics, 2007, 71:1, 53–56

Bibliographic databases:

UDC: 512.817.72+512.817.6
MSC: 20F55, 51F15, 22E40
Received: 14.09.2006

Citation: V. V. Nikulin, “Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces”, Izv. RAN. Ser. Mat., 71:1 (2007), 55–60; Izv. Math., 71:1 (2007), 53–56

Citation in format AMSBIB
\by V.~V.~Nikulin
\paper Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces
\jour Izv. RAN. Ser. Mat.
\yr 2007
\vol 71
\issue 1
\pages 55--60
\jour Izv. Math.
\yr 2007
\vol 71
\issue 1
\pages 53--56

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    This publication is cited in the following articles:
    1. Agol I., Belolipetsky M., Storm P., Whyte K., “Finiteness of arithmetic hyperbolic reflection groups”, Groups Geom. Dyn., 2:4 (2008), 481–498  crossref  mathscinet  zmath  isi
    2. V. V. Nikulin, “On Ground Fields of Arithmetic Hyperbolic Reflection Groups. II”, Mosc. Math. J., 8:4 (2008), 789–812  mathnet  crossref  mathscinet  zmath
    3. Nikulin V.V., “On ground fields of arithmetic hyperbolic reflection groups. III”, J. Lond. Math. Soc. (2), 79:3 (2009), 738–756  crossref  mathscinet  zmath  isi  scopus
    4. Nikulin V.V., “On ground fields of arithmetic hyperbolic reflection groups”, Groups and symmetries, CRM Proc. Lecture Notes, 47, Amer. Math. Soc., Providence, RI, 2009, 299–326  crossref  mathscinet  zmath  isi
    5. Maclachlan C., “Bounds for discrete hyperbolic arithmetic reflection groups in dimension 2”, Bull. Lond. Math. Soc., 43:1 (2011), 111–123  crossref  mathscinet  zmath  isi  elib  scopus
    6. 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
    7. Belolipetsky M., “Finiteness theorems for congruence reflection groups”, Transform. Groups, 16:4 (2011), 939–954  crossref  mathscinet  zmath  isi  scopus
    8. Maclachlan C., “Commensurability classes of discrete arithmetic hyperbolic groups”, Groups Geom. Dyn., 5:4 (2011), 767–785  crossref  mathscinet  zmath  isi  scopus
    9. Mcleod J., “Hyperbolic reflection groups associated to the quadratic forms $-3x^2_0+x^2_1+…+x^2_n$”, Geom. Dedicata, 152:1 (2011), 1–16  crossref  mathscinet  zmath  isi  scopus
    10. M. Belolipetsky, B. Linowitz, “On Fields of Definition of Arithmetic Kleinian Reflection Groups II”, International Mathematics Research Notices, 2013  crossref  mathscinet  scopus
    11. Anna Felikson, Pavel Tumarkin, “Essential hyperbolic Coxeter polytopes”, Isr. J. Math, 2013  crossref  mathscinet  scopus
    12. Mark A., “Reflection Groups of the Quadratic Form -Px(0)(2) + X(1)(2) + ... X(N)(2) With P Prime”, Publ. Mat., 59:2 (2015), 353–372  crossref  mathscinet  zmath  isi  scopus
    13. Belolipetsky M., “Arithmetic hyperbolic reflection groups”, Bull. Amer. Math. Soc., 53:3 (2016), 437–475  crossref  mathscinet  zmath  isi  elib  scopus
    14. Ishida M., “Cusp Singularities and Quasi-Polyhedral Sets”, Algebraic Varieties and Automorphism Groups, Advanced Studies in Pure Mathematics, 75, eds. Masuda K., Kishimoto T., Kojima H., Miyanishi M., Zaidenberg M., Math Soc Japan, 2017, 163–182  mathscinet  zmath  isi
    15. 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  mathscinet  zmath  isi  scopus
    16. Linowitz B., “Bounds For Arithmetic Hyperbolic Reflection Groups in Dimension 2”, Transform. Groups, 23:3 (2018), 743–753  crossref  mathscinet  isi  scopus
    17. N. V. Bogachev, A. Yu. Perepechko, “Vinberg's Algorithm for Hyperbolic Lattices”, Math. Notes, 103:5 (2018), 836–840  mathnet  crossref  crossref  isi  elib
    18. 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
    19. Kontorovich A., Nakamura K., “Geometry and Arithmetic of Crystallographic Sphere Packings”, Proc. Natl. Acad. Sci. U. S. A., 116:2 (2019), 436–441  crossref  isi  scopus
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