Abstract
The invariant trace field and quaternion algebra of a finite-covolume Kleinian group was introduced in Chapter 3 accompanied by methods to enable the computation of these invariants to be made. Such computations were carried out in Chapter 4 for a variety of examples. We now consider some general applications of these invariants to problems in the geometry and topology of hyperbolic 3-manifolds. Generally, these have the form that special properties of the invariants have geometric consequences for the related manifolds or groups. In some cases, to fully exploit these applications, the existence of manifolds or groups whose related invariants have these special properties requires the construction of arithmetic Kleinian groups, and these cases will be revisited in later chapters.
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
Takeuchi, K. (1975). A characterization of arithmetic Fuchsian groups. J. Math. Soc. Japan, 27: 600–612.
Reid, A. (1987). Arithmetic Kleinian groups and their Fuchsian subgroups. PhD thesis, Aberdeen University.
Gehring, F., Maclachlan, C., Martin, G., and Reid, A. (1997). Arithmet- icity, discreteness and volume. Trans. Am. Math. Soc, 349: 3611–3643.
Maclachlan, C. and Rosenberger, G. (1983). Two-generator arithmetic Fuchsian groups. Math. Proc. Cambridge Phil. Soc, 93: 383–391.
Morgan, J. and Bass, H., editors (1984). The Smith Conjecture. Academic Press, Orlando, FL.
Bass, H. (1980). Groups of integral representation type. Pacific J. Math, 86: 15–51.
Culler, M., Gordon, C., Luecke, J., and Shalen, P. (1987). Dehn surgery on knots. Annals of Math, 125: 237–300.
Serre, J.-P. (1980). Trees. Springer-Verlag, Berlin.
Thurston, W. (1979). The geometry and topology of three-manifolds. Notes from Princeton University.
Jones, K. and Reid, A. (1998). Minimal index torsion free subgroups of Kleinian groups. Math. Annalen, 310: 235–250.
Vinberg, E. (1985). Hyperbolic reflection groups. Russian Math. Surveys, 40: 31–75.
Reid, A. (1991b). Totally geodesic surfaces in hyperbolic 3-manifolds. Proc. Edinburgh Math. Soc, 34: 77–88.
Scott, P. (1978). Subgroups of surface groups are almost geometric. J. London Math. Soc, 17: 555–565.
Riley, R. (1974). Knots with parabolic property P. Quart. J. Math, 25: 273–283.
Riley, R. (1972). Parabolic representations of knot groups, I. Proc. London Math. Soc, 24: 217–247.
Hoste, J. and Shanahan, P. (2001). Trace fields of twist knots. J. Knot Theory and its Ramifications, 10: 625–639.
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). Applications. 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_6
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6720-9_6
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