## Computing arithmetic Kleinian groups

HTML articles powered by AMS MathViewer

- by Aurel Page PDF
- Math. Comp.
**84**(2015), 2361-2390 Request permission

## 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.## References

- Wieb Bosma, John Cannon, and Catherine Playoust,
*The Magma algebra system. I. The user language*, J. Symbolic Comput.**24**(1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR**1484478**, DOI 10.1006/jsco.1996.0125 - 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**, DOI 10.4007/annals.2010.172.2197 - 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** - 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**, DOI 10.1017/S1474748012000667 - 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**, DOI 10.1090/surv/067 - 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** - 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**, DOI 10.1016/j.aim.2003.07.015 - Frank Calegari and Akshay Venkatesh,
*A torsion Jacquet–Langlands correspondence*, http://arxiv.org/abs/1212.3847. - 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**, DOI 10.1090/S0025-5718-1985-0777278-8 - H. Jacquet and R. P. Langlands,
*Automorphic forms on $\textrm {GL}(2)$*, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR**0401654** - 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. - M. Lipyanskiy,
*A computer-assisted application of Poincaré’s fundamental polyhedron theorem*, Preprint available at http://www.math.columbia.edu/~ums/Archive.html, 2002. - Bernard Maskit,
*On Poincaré’s theorem for fundamental polygons*, Advances in Math.**7**(1971), 219–230. MR**297997**, DOI 10.1016/S0001-8708(71)80003-8 - 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**, DOI 10.1007/978-1-4757-6720-9 - A. Page,
*Computing fundamental domains for arithmetic Kleinian groups*, Master’s thesis, Université Paris 7, August 2010. - 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. - John G. Ratcliffe,
*Foundations of hyperbolic manifolds*, 2nd ed., Graduate Texts in Mathematics, vol. 149, Springer, New York, 2006. MR**2249478** - Robert Riley,
*Applications of a computer implementation of Poincaré’s theorem on fundamental polyhedra*, Math. Comp.**40**(1983), no. 162, 607–632. MR**689477**, DOI 10.1090/S0025-5718-1983-0689477-2 - Richard G. Swan,
*Generators and relations for certain special linear groups*, Advances in Math.**6**(1971), 1–77 (1971). MR**284516**, DOI 10.1016/0001-8708(71)90027-2 - The PARI Group, Bordeaux,
*PARI/GP, version 2.6.0*, 2011, available from http://pari.math.u-bordeaux.fr/. - Marie-France Vignéras,
*Arithmétique des algèbres de quaternions*, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR**580949** - 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** - 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**, DOI 10.1007/978-3-642-14518-6_{3}0

## 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
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3356030