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Applications

  • Colin Maclachlan
  • Alan W. Reid
Part of the Graduate Texts in Mathematics book series (GTM, volume 219)

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.

Keywords

Kleinian Group Fuchsian Group Quaternion Algebra Algebraic Integer Incompressible Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Further Reading

  1. Takeuchi, K. (1975). A characterization of arithmetic Fuchsian groups. J. Math. Soc. Japan, 27: 600–612.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Reid, A. (1987). Arithmetic Kleinian groups and their Fuchsian subgroups. PhD thesis, Aberdeen University.Google Scholar
  3. Gehring, F., Maclachlan, C., Martin, G., and Reid, A. (1997). Arithmet- icity, discreteness and volume. Trans. Am. Math. Soc, 349: 3611–3643.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Maclachlan, C. and Rosenberger, G. (1983). Two-generator arithmetic Fuchsian groups. Math. Proc. Cambridge Phil. Soc, 93: 383–391.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Morgan, J. and Bass, H., editors (1984). The Smith Conjecture. Academic Press, Orlando, FL.zbMATHGoogle Scholar
  6. Bass, H. (1980). Groups of integral representation type. Pacific J. Math, 86: 15–51.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Culler, M., Gordon, C., Luecke, J., and Shalen, P. (1987). Dehn surgery on knots. Annals of Math, 125: 237–300.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Serre, J.-P. (1980). Trees. Springer-Verlag, Berlin.zbMATHCrossRefGoogle Scholar
  9. Thurston, W. (1979). The geometry and topology of three-manifolds. Notes from Princeton University.Google Scholar
  10. Jones, K. and Reid, A. (1998). Minimal index torsion free subgroups of Kleinian groups. Math. Annalen, 310: 235–250.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Vinberg, E. (1985). Hyperbolic reflection groups. Russian Math. Surveys, 40: 31–75.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Reid, A. (1991b). Totally geodesic surfaces in hyperbolic 3-manifolds. Proc. Edinburgh Math. Soc, 34: 77–88.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Scott, P. (1978). Subgroups of surface groups are almost geometric. J. London Math. Soc, 17: 555–565.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Riley, R. (1974). Knots with parabolic property P. Quart. J. Math, 25: 273–283.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Riley, R. (1972). Parabolic representations of knot groups, I. Proc. London Math. Soc, 24: 217–247.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Hoste, J. and Shanahan, P. (2001). Trace fields of twist knots. J. Knot Theory and its Ramifications, 10: 625–639.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Colin Maclachlan
    • 1
  • Alan W. Reid
    • 2
  1. 1.Department of Mathematical Sciences, Kings CollegeUniversity of AberdeenAberdeenUK
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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