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Finiteness theorems for congruence reflection groups

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Abstract

This paper is a follow-up to the paper I. Agol, M. Belolipetsky, P. Storm, K. Whyte, Finiteness of arithmetic hyperbolic reflection groups, Groups, Geometry, and Dynamics 2 (2008), 481–498. The main purpose is to investigate the effective side of the method developed there and its possible application to the problem of classification of arithmetic hyperbolic reection groups.

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Further Reading

  1. I. Agol, Finiteness of arithmetic Kleinian reection groups, in: Proceedings of the International Congress of Mathematicians, Madrid, 2006, Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 951–960.

  2. I. Agol, M. Belolipetsky, P. Storm, K. Whyte, Finiteness of arithmetic hyperbolic reflection groups, Groups, Geometry, and Dynamics 2 (2008), 481–498.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Allcock, Infinitely many hyperbolic Coxeter groups through dimension 19, Geom. Topology 10 (2006), 737–758.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Belolipetsky, On fields of definition of arithmetic Kleinian reection groups, Proc. Amer. Math. Soc. 137 (2009), 1035–1038.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Belolipetsky, On volumes of arithmetic quotients of SO(1, n), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3 (2004), 749–770; Addendum: ibid. 6 (2007), 263–268.

    MathSciNet  MATH  Google Scholar 

  6. M. Belolipetsky, Counting maximal arithmetic subgroups (with an appendix by J. Ellenberg and A. Venkatesh), Duke Math. J. 140 (2007), no. 1, 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Belolipetsky, V. Emery, On volumes of arithmetic quotients of PO(n, 1)°, n odd, arXiv:1001.4670v1.

  8. M. Belolipetsky, W. T. Gan, The mass of unimodular lattices, J. Number Theory 114 (2005), 221–237.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Belolipetsky, T. Gelander, A. Lubotzky, A. Shalev, Counting arithmetic lattices and surfaces, Ann. of Math. 172 (2010), 2197–2221.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), 133–153.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Borel, Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), 1–33.

    MathSciNet  MATH  Google Scholar 

  12. A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 119–171; Addendum: ibid. 71 (1990), 173–177.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Bozejko, T. Januszkiewicz, R. Spatzier, Infinite Coxeter groups do not have Kazhdan's property, J. Operator Theory 19 (1988), 63–67.

    MathSciNet  MATH  Google Scholar 

  14. R. Brooks, Some relations between spectral geometry and number theory, in: Topology’ 90 (Columbus, OH, 1990), Ohio State Univ. Math. Res. Inst. Publ., Vol. 1, de Gruyter, Berlin, 1992, pp. 61–75.

  15. V. O. Bugaenko, On reflective unimodular hyperbolic quadratic forms, Selecta Math. Soviet. 9 (1990) 263–271.

    MathSciNet  MATH  Google Scholar 

  16. M. Burger, P. Sarnak, Ramanujan duals, II, Invent. Math. 106 (1991), 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  17. T. Chinburg, E. Friedman, The smallest arithmetic hyperbolic three-orbifold, Invent. Math. 86 (1986), 507–527.

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Deligne, G. D. Mostow, Commensurabilities among lattices in PU(1, n), Annals of Mathematics Studies, Vol. 132, Princeton Univ. Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  19. V. Emery, Du volume des quotients arithmétiques de l'espace hyperbolique, Ph.D. thesis no. 1648, University of Fribourg, 2009.

  20. W. T. Gan, J. Hanke, J.-K. Yu, On an exact mass formula of Shimura, Duke Math. J. 107 (2001), 103–133.

    Article  MathSciNet  MATH  Google Scholar 

  21. Д. А. Каждан, О связц дуального пространства группы со строенцем её замкнутых подгрупп, Функц. анализ и его прил. 1 (1967), no. 1, 71–74. Engl. transl.: D. Kazhdan, On the connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal. Appl. 1 (1967), 63–65.

    Google Scholar 

  22. J.-S. Li, On the first eigenvalue of Laplacian for locally symmetric manifolds, in: First International Congress of Chinese Mathematicians (Beijing, 1998), AMS/IP Stud. Adv. Math., Vol. 20, Amer. Math. Soc., Providence, RI, 2001, pp. 271–278.

  23. P. Li, S. T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), 269–291.

    Article  MathSciNet  MATH  Google Scholar 

  24. D. D. Long, C. Maclachlan, A. W. Reid, Arithmetic Fuchsian groups of genus zero, Pure Appl. Math. Quart. 2 (2006), no. 2, 1–31.

    MathSciNet  Google Scholar 

  25. C. Maclachlan, Bounds for discrete hyperbolic arithmetic reflection groups in dimension 2, preprint, 2009.

  26. G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 17, Springer-Verlag, Berlin, 1991. Russian transl.: Г. А. Маргулис, Дцскретные подгруппы групп Лц, Лц, Изд-во МЦНМО, М., 2007.

    MATH  Google Scholar 

  27. J. Mcleod, Hyperbolic reection groups associated to the quadratic forms -3x 20  + x 21  + ⋯ + x 2 n , Geom. Dedicata 152 (2011), 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. Meyerhoff, Sphere-packing and volume in hyperbolic 3-space, Comment. Math. Helv. 61 (1986), 271–278.

    Article  MathSciNet  MATH  Google Scholar 

  29. O. O’Meara, Introduction to Quadratic Forms, Die Grundlehren der mathematischen Wissenschaften, Vol. 117, Academic Press, New York, Springer-Verlag, Berlin, 1963.

    MATH  Google Scholar 

  30. В. В. Никулин, Конечность чцсла арцфметцческцх групп, поржсдённых отражсенцямц, в пространствах Лобачевского, Изв. РАН, Сер. мат. 71 (2007), no. 1, 55–60. Engl. transl.: V. Nikulin, Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces, Izvestiya: Math. 71 (2007), no. 1, 53–56.

  31. V. Nikulin, The transition constant for arithmetic hyperbolic reflection groups, arXiv:0910.5217v4.

  32. A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: A survey of recent results, Sémin. Théor. Nombres Bordeaux, Sér. II 2 (1990), 119–141.

  33. A. M. Odlyzko, Discriminant bounds, http://www.dtc.umn.edu/_odlyzko/unpub-lished/index.html.

  34. В. П. Платонов, А. С. Рапинчук, Алгебрацческце группы ц теорця чцсел, Наука, М., 1991. Engl. transl.: V. P. Platonov, A. S. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Mathematics, Vol. 139, Academic Press, Boston, MA, 1994.

  35. G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 91–117.

    Article  MATH  Google Scholar 

  36. C. L. Siegel, Some remarks on discontinuous groups, Ann. of Math. 46 (1945), 708–718.

    Article  MathSciNet  MATH  Google Scholar 

  37. M.-F. Vignéras, Quelques remarques sur la conjecture λ1 ≥ 1/4, in: Seminar on Number Theory, Paris 1981–82 (Paris, 1981/1982), Progress in Mathematics, Vol. 38, Birkhäuser Boston, Boston, MA, 1983, pp. 321–343.

  38. Э. Б. Винберг, Лцскретные группы, порождённые отраженцямц, в пространстыах Лобачевского, Мат. сб. 72 (114) (1967), no. 3, 471–488; исправление, ibid. 73 (115) (1967), no. 2, 303. Engl. transl.: È. B. Vinberg, Discrete groups generated by reections in Lobachevskii spaces, Math. USSR-Sb. 1 (1967), 429–444.

  39. Э. Б. Винберг, О группах едцнцц некоторых квадратцчных форм, Мат. сб. 87(129) (1972), 18–36. Engl. transl.: È. B. Vinberg, On groups of unit elements of certain quadratic forms, Math. USSR, Sb. 16 (1972), no. 1, 17–35.

  40. Э. Б. Винберг, Отсутствце крцсталлграфцческцх групп отраженцй в простраиствах Лобачвского большой размерностц, Функц. анализ и его прил. 15 (1981), no. 2, 67–68. Engl. transl.: È. B. Vinberg, Absence of crystallographic groups of reflections in Lobachevskij spaces of large dimension, Funct. Anal. Appl. 15 (1981), 128–130.

  41. Э. Б. Винберг, Отсутствце крцсталлграфцческцх групп отраженцй в простраиствах Лобачвского большой размерностц, Труды ММО 47 (1984), 68–102, 246. Engl. transl.: È. B. Vinberg, The nonexitence of crystallographic groups of reflections in Lobachevskii spaces of large dimension, Trans. Mosc. Math. Soc. (1985), 75–112.

  42. È. B. Vinberg, Some free algebras of automorphic forms on symmetric domains of type IV, Transform. Groups 15 (2010), 701–741.

    Article  MathSciNet  MATH  Google Scholar 

  43. Э. Б. Винберг, И. М. Каплинская, Группы O 18,1 (\( \mathbb{Z} \)) u O 19,1 (\( \mathbb{Z} \)), ДАН CCCP 238 (1978), 1273–1275. Engl. transl.: È. B. Vinberg, I. M. Kaplinskaja, The groups O 18,1 (\( \mathbb{Z} \)) and O 19,1 (\( \mathbb{Z} \)), Sov. Math., Dokl. 19 (1978), 194–197.

  44. H. C. Wang, Topics on totally discontinuous groups, in: Symmetric Spaces, (Short Courses, Washington Univ., St. Louis, Mo., 1969{1970), Pure and Appl. Math., Vol. 8, Dekker, New York, 1972, pp. 459–487.

  45. The Bordeaux Database, ftp://megrez.math.u-bordeaux.fr/pub/numberfields/.

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Correspondence to Mikhail Belolipetsky.

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Partially supported by EPSRC grant EP/F022662/1.

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Belolipetsky, M. Finiteness theorems for congruence reflection groups. Transformation Groups 16, 939–954 (2011). https://doi.org/10.1007/s00031-011-9156-3

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