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Computing arithmetic Kleinian groups


Author: Aurel Page
Journal: Math. Comp. 84 (2015), 2361-2390
MSC (2010): Primary 11F06, 11Y99, 30F40; Secondary 11-04, 16U60, 11R52
DOI: https://doi.org/10.1090/S0025-5718-2015-02939-1
Published electronically: March 13, 2015
MathSciNet review: 3356030
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Abstract | References | Similar Articles | Additional Information

Abstract: Arithmetic Kleinian groups are arithmetic lattices in  $ \mathrm {PSL}_2(\mathbb{C})$. We present an algorithm that, given such a group $ \Gamma $, returns a fundamental domain and a finite presentation for $ \Gamma $ with a computable isomorphism.


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  • [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, Computational Algebra and Number Theory (London, 1993), J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. MR 1484478, https://doi.org/10.1006/jsco.1996.0125
  • [BGLS10] Mikhail Belolipetsky, Tsachik Gelander, Alexander Lubotzky, and Aner Shalev, Counting arithmetic lattices and surfaces, Ann. of Math. (2) 172 (2010), no. 3, 2197-2221. MR 2726109 (2011i:11150), https://doi.org/10.4007/annals.2010.172.2197
  • [Bor81] A. Borel, Commensurability classes and volumes of hyperbolic $ 3$-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 1-33. MR 616899 (82j:22008)
  • [BV13] Nicolas Bergeron and Akshay Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013), no. 2, 391-447. MR 3028790, https://doi.org/10.1017/S1474748012000667
  • [BW00] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR 1721403 (2000j:22015)
  • [CFJR01] Ted Chinburg, Eduardo Friedman, Kerry N. Jones, and Alan W. Reid, The arithmetic hyperbolic 3-manifold of smallest volume, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 1, 1-40. MR 1882023 (2003a:57027)
  • [CJLdR04] Capi Corrales, Eric Jespers, Guilherme Leal, and Angel del Río, Presentations of the unit group of an order in a non-split quaternion algebra, Adv. Math. 186 (2004), no. 2, 498-524. MR 2073916 (2005d:16051), https://doi.org/10.1016/j.aim.2003.07.015
  • [CV12] Frank Calegari and Akshay Venkatesh, A torsion Jacquet-Langlands correspondence, http://arxiv.org/abs/1212.3847.
  • [FP85] U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comp. 44 (1985), no. 170, 463-471. MR 777278 (86e:11050), https://doi.org/10.2307/2007966
  • [JL70] H. Jacquet and R. P. Langlands, Automorphic Forms on $ {\rm GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin, 1970. MR 0401654 (53 #5481)
  • [Kan83] R. Kannan, Improved algorithms for integer programming and related lattice problems, Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing (New York, NY, USA), STOC '83, ACM, 1983, pp. 193-206.
  • [Lip02] M. Lipyanskiy, A computer-assisted application of Poincaré's fundamental polyhedron theorem, Preprint available at http://www.math.columbia.edu/~ums/Archive.html, 2002.
  • [Mas71] Bernard Maskit, On Poincaré's theorem for fundamental polygons, Advances in Math. 7 (1971), 219-230. MR 0297997 (45 #7049)
  • [MR03] Colin Maclachlan and Alan W. Reid, The Arithmetic of Hyperbolic 3-Manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR 1937957 (2004i:57021)
  • [Pag10] A. Page, Computing fundamental domains for arithmetic Kleinian groups, Master's thesis, Université Paris 7, August 2010.
  • [Rah10] A. Rahm, (Co)homologies et K-théorie de groupes de Bianchi par des modèles géométriques calculatoires, PhD thesis, Université Joseph-Fourier - Grenoble I, October 2010.
  • [Rat06] John G. Ratcliffe, Foundations of Hyperbolic Manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 149, Springer, New York, 2006. MR 2249478 (2007d:57029)
  • [Ril83] Robert Riley, Applications of a computer implementation of Poincaré's theorem on fundamental polyhedra, Math. Comp. 40 (1983), no. 162, 607-632. MR 689477 (85b:20064), https://doi.org/10.2307/2007537
  • [Swa71] Richard G. Swan, Generators and relations for certain special linear groups, Advances in Math. 6 (1971), 1-77 (1971). MR 0284516 (44 #1741)
  • [The11] The PARI Group, Bordeaux, PARI/GP, version 2.6.0, 2011, available from http://pari.math.u-bordeaux.fr/.
  • [Vig80] Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949 (82i:12016)
  • [Voi09] John Voight, Computing fundamental domains for Fuchsian groups, J. Théor. Nombres Bordeaux 21 (2009), no. 2, 469-491 (English, with English and French summaries). MR 2541438 (2011c:11064)
  • [Yas10] Dan Yasaki, Hyperbolic tessellations associated to Bianchi groups, Algorithmic Number Theory, Lecture Notes in Comput. Sci., vol. 6197, Springer, Berlin, 2010, pp. 385-396. MR 2721434 (2012g:11069), https://doi.org/10.1007/978-3-642-14518-6_30

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Additional Information

Aurel Page
Affiliation: Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France, CNRS, IMB, UMR 5251, F-33400 Talence, France, INRIA, F-33400 Talence, France
Email: aurel.page@math.u-bordeaux1.fr; a.r.page@warwick.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-2015-02939-1
Received by editor(s): May 31, 2012
Received by editor(s) in revised form: February 6, 2013, September 20, 2013, and February 2, 2014
Published electronically: March 13, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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