Abstract
In the preceding chapter, arithmetic Kleinian groups were defined and identified amongst all Kleinian groups. Thus several examples from earlier chapters can be reassessed as being arithmetic, thus enhancing their study. Moreover, the existence part of the classification theorem for quaternion algebras (Theorem 7.3.6) gives the existence of arithmetic Kleinian groups satisfying a variety of conditions, which, in turn, give the existence of hyperbolic 3-manifolds and orbifolds with a range of topological and geometric properties. These aspects will be explored in this chapter.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Further Reading
Goldstein, L. (1971). Analytic Number Theory. Prentice-Hall, Englewood, NJ.
Hempel, J. (1986). Coverings of Dehn fillings on surface bundles. Topology and its Applications, 24: 157–170.
Lam, T. (1973). The Algebraic Theory of Quadratic Forms. W.A. Benjamin, Reading, MA.
Grunewald, F. and Schwermer, J. (1981b). Free non-abelian quotients of SL2 over orders in imaginary quadratic number fields. J. Algebra, 69: 298–304.
Lang, S. (1970). Algebraic Number Theory. Addison-Wesley, Reading, MA.
Lubotzky, A. (1982). Free quotients and the congruence kernel of SL2. J. of Algebra, 77: 411–418.
Grunewald, F. and Schwermer, J. (1981b). Free non-abelian quotients of SL2 over orders in imaginary quadratic number fields. J. Algebra, 69: 298–304.
Maclachlan, C. and Reid, A. (1991). Parametrizing Fuchsian subgroups of the Bianchi groups. Canadian J. Math, 43: 158–181.
Pohst, M. and Zassenhaus, H. (1989). Algorithmic Algebraic Number Theory. Cambridge University Press, Cambridge.
Culler, M., Hersonsky, S., and Shalen, P. (1998). The first Betti number of the smallest hyperbolic 3-manifold. Topology, 37: 807–849.
Reiter, H. (1968). Classical Harmonic Analysis and Locally Compact Groups. Oxford University Press, Oxford.
Riley, R. (1979). An elliptical path from parabolic representations to hyperbolic structures, pages 99–133. Lecture Notes in Mathematics No. 722. Springer-Verlag, Heidelberg.
Serre, J.-P. (1964). Cohomologie Galoisienne. Lecture Notes in Mathematics No. 5. Springer-Verlag, Berlin.
Anderson, J. (1999). Hyperbolic Geometry. Springer-Verlag, New York.
Beardon, A. (1983). The Geometry of Discrete Groups. Graduate Texts in Mathematics Vol. 91. Springer-Verlag, New York.
Hasse, H. (1980). Number Theory. Grundlehren der mathematischen Wissenschaften, Vol. 229. Springer-Verlag, Berlin.
Neumann, W. and Reid, A. (1991). Amalgamation and the invariant trace field of a Kleinian group. Math. Proc. Cambridge Phil. Soc, 109: 509–515.
Reid, A. (1991a). Arithmeticity of knot complements. J. London Math. Soc, 43: 171–184.
Reid, A. (1991b). Totally geodesic surfaces in hyperbolic 3-manifolds. Proc. Edinburgh Math. Soc, 34: 77–88.
Thomas, R. (1991). The Fibonacci groups revisited, pages 445–456. L. M. S. Lecture Notes Series Vol. 160. Cambridge University Press, Cambridge.
Hasse, H. (1980). Number Theory. Grundlehren der mathematischen Wissenschaften, Vol. 229. Springer-Verlag, Berlin.
Grunewald, F. and Schwermer, J. (1981c). A non-vanishing result for the cuspidal cohomology of SL(2) over imaginary quadratic integers. Math. Annalen, 258: 183–200.
Grunewald, F. and Schwermer, J. (1981a). Arithmetic quotients of hyperbolic 3-space, cusp forms and link complements. Duke Math. J, 48: 351–358.
Haas, A. (1985). Length spectra as moduli for hyperbolic surfaces. Duke Math. J, 52: 923–934.
Helling, H., Kim, A., and Mennicke, J. (1998). A geometric study of Fibonacci groups. J. Lie Theory, 8: 1–23.
Jones, K. and Reid, A. (1998). Minimal index torsion free subgroups of Kleinian groups. Math. Annalen, 310: 235–250.
Matsuzaki, K. and Taniguchi, M. (1998). Hyperbolic Manifolds and Kleinian Groups. Oxford University Press, Oxford.
Reid, A. (1987). Arithmetic Kleinian groups and their Fuchsian subgroups. PhD thesis, Aberdeen University.
Ruberman, D. (1987). Mutation and volumes of links. Invent. Math, 90: 189–215.
Stewart, I. and Tall, D. (1987). Algebraic Number Theory Chapman and Hall, London.
Adams, C. (1987). The non-compact hyperbolic 3-manifold of minimum volume. Proc. Am. Math. Soc, 100: 601–606.
Baumslag, G., Morgan, J., and Shalen, P. (1987). Generalized triangle groups. Math. Proc. Cambridge Phil. Soc, 102: 25–31.
Brooks, R. and Tse, R. (1987). Isospectral surfaces of small genus. Nagoya Math. J, 107: 13–24.
Chinburg, T. (1987). A small arithmetic hyperbolic 3-manifold. Proc. Am. Math. Soc, 100: 140–144.
Hejhal, D. (1976). The Selberg Trace Formula for PSL(2,R). Volume 1. Lecture Notes in Mathematics No. 548. Springer-Verlag, Berlin.
Bleiler, S. and Hodgson, C. (1996). Spherical space forms and Dehn filling. Topology, 35: 809–833.
Dobrowolski, E. (1979). On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith, 34: 391–401.
Cohn, H. (1978). A Classical Invitation to Algebraic Numbers and Class Fields. Springer-Verlag, New York.
Beardon, A. (1983). The Geometry of Discrete Groups. Graduate Texts in Mathematics Vol. 91. Springer-Verlag, New York.
Cremona, J. (1984). Hyperbolic tesselations, modular symbols and elliptic curves over complex quadratic fields. Compos. Math, 51: 275–323.
Adams, C. (1992). Noncompact hyperbolic orbifolds of small volume. In Topology ‘80,pages 1–15, Berlin. de Gruyter.
Bleiler, S. and Hodgson, C. (1996). Spherical space forms and Dehn filling. Topology, 35: 809–833.
Baker, M. (2001). Link complements and the Bianchi modular groups. Trans. Am. Math. Soc, 353: 3229–3246.
Baker, M. and Reid, A. (2002). Arithmetic knots in closed 3-manifolds. J. Knot Theory and its Ramifications, 11.
Bart, A. (2001). Surface groups in some surgered manifolds. Topology, 40: 197–211.
Bleiler, S. and Hodgson, C. (1996). Spherical space forms and Dehn filling. Topology, 35: 809–833.
Cooper, D. and Long, D. (1997). The A-polynomial has ones in the corners. Bull. London Math. Soc, 29: 231–238.
Chinburg, T. and Reid, A. (1993). Closed hyperbolic 3-manifolds whose closed geodesics all are simple. J. Differ. Geom, 38: 545–558.
Chinburg, T. and Friedman, E. (2000). The finite subgroups of maximal arithmetic subgroups of PGL(2, C). Ann. Inst. Fourier, 50: 1765–1798.
Cooper, D. and Long, D. (2001). Some surface groups survive surgery. Geom. and Topol, 5: 347–367.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Maclachlan, C., Reid, A.W. (2003). Arithmetic Hyperbolic 3-Manifolds and Orbifolds. In: The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Mathematics, vol 219. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-6720-9_10
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6720-9_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-3122-1
Online ISBN: 978-1-4757-6720-9
eBook Packages: Springer Book Archive