K. That-for-every, G. /. {h, and . H}, We write the projection ?, ) ? G be a lifting)) ? G. We write the projection e : F (S(H)) H

H. ?? and . ??-g, Suppose we have an exact sequence of groups 1 ??

. Suppose, K. For, and G. /. {h, Then the set S(H) ? S(G) is a set of generators for G and we have a finite presentation

R. Bosma, J. J. Cannon, and C. Playoust, The Magma Algebra System I: The User Language, Journal of Symbolic Computation, vol.24, issue.3-4, pp.235-265, 1997.
DOI : 10.1006/jsco.1996.0125

]. H. Boe05 and . Boehm, The constructive reals as a Java library, Journal of Logic and Algebraic Programming, vol.64, issue.1, pp.3-11, 2005.

]. C. Cjldr04, E. Corrales, G. Jespers, and . Leal, Presentations of the unit group of an order in a non-split quaternion algebra* 1, Advances in Mathematics, vol.186, issue.2, pp.498-524, 2004.

M. [. Fincke and . Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Mathematics of Computation, vol.44, issue.170, pp.463-471, 1985.
DOI : 10.1090/S0025-5718-1985-0777278-8

D. [. Fieker and . Stehlé, Short Bases of Lattices over Number Fields, Proceedings of ANTS-IX, 2006.
DOI : 10.1007/978-3-642-14518-6_15
URL : https://hal.archives-ouvertes.fr/hal-00546895

D. [. Gowland and . Lester, A Survey of Exact Arithmetic Implementations, Computability and Complexity in Analysis, pp.30-47, 2001.
DOI : 10.1007/3-540-45335-0_3

]. S. Kat92 and . Katok, Fuchsian groups, 1992.

]. A. Mar07 and . Marden, Outer Circles: An introduction to hyperbolic 3- manifolds, 2007.

]. B. Mas71 and . Maskit, On Poincaré's theorem for fundamental polygons, Advances in Mathematics, vol.7, issue.3, pp.219-230, 1971.

]. B. Mas88 and . Maskit, Kleinian groups, 1988.

A. [. Maclachlan and . Reid, The arithmetic of hyperbolic 3- manifolds, 2003.
DOI : 10.1007/978-1-4757-6720-9

J. [. Pour-el and . Richards, Computability in analysis and physics, 1989.

]. J. [-rat06 and . Ratcliffe, Foundations of hyperbolic manifolds, 2006.

]. R. Ril83 and . Riley, Applications of a computer implementation of Poincaré's theorem on fundamental polyhedra, Mathematics of Computation, vol.40, issue.162, pp.607-632, 1983.

M. [. Schnorr and . Euchner, Lattice basis reduction: Improved practical algorithms and solving subset sum problems, Mathematical Programming, vol.13, issue.1, pp.181-199, 1994.
DOI : 10.1007/BF01581144

]. J. Ser73 and . Serre, A course in arithmetic, 1973.

H. [. Schnorr and . Hoerner, Attacking the Chor-Rivest Cryptosystem by Improved Lattice Reduction, Proceedings of the 14th annual international conference on Theory and application of cryptographic techniques, pp.1-12, 1995.
DOI : 10.1007/3-540-49264-X_1

]. R. Swa71 and . Swan, Generators and relations for certain special linear groups, Advances in Math, vol.6, pp.1-771971, 1971.

]. M. Vig80 and . Vignéras, Arithmétique des algèbres de quaternions, 1980.

]. J. Voi09 and . Voight, Computing fundamental domains for Fuchsian groups, pp.467-489, 2009.

]. J. Voi10 and . Voight, Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms, 2010.

]. K. Wei00 and . Weihrauch, Computable analysis: an introduction, 2000.

]. D. Yas09 and . Yasaki, Hyperbolic tessellations associated to Bianchi groups Arxiv preprint, 2009.