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


In this chapter, the invariant trace fields and quaternion algebras of a number of classical examples of hyperbolic 3-manifolds and Kleinian groups will be determined Many of these will be considered again in greater detail later, to illustrate certain applications or to extract more information on the manifolds or orbifolds, particularly in the cases where the groups turn out to be arithmetic. However, already in this chapter, these examples will exhibit certain properties which answer some basic questions on hyperbolic 3-orbifolds and manifolds. Stronger applications of the invariance will be made in the next chapter. For the moment, we will illustrate the results and methods of the preceding chapter by calculating the invariant trace fields and quaternion algebras of some familiar examples. The methods exhibited by these examples should enable the reader to carry out the determination of the invariant trace field and quaternion algebra of the particular favourite example in which they are interested.


Kleinian Group Fuchsian Group Quaternion Algebra Algebraic Integer Parabolic Element 
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Further Reading

  1. Hatcher, A. (1983). Hyperbolic structures of arithmetic type on some link complements. J. London Math. Soc, 27: 345–355.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Hejhal, D. (1976). The Selberg Trace Formula for PSL(2,R). Volume 1. Lecture Notes in Mathematics No. 548. Springer-Verlag, Berlin.Google Scholar
  3. Swan, R. (1971). Generators and relations for certain special linear groups. Adv. Math, 6: 1–77.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Fine, B. (1989). Algebraic Theory of the Bianchi Groups. Marcel Dekker, New York.zbMATHGoogle Scholar
  5. Elstrodt, J., Grunewald, F., and Mennicke, J. (1998). Groups Acting on Hyperbolic Space. Monographs in Mathematics. Springer-Verlag, Berlin.zbMATHCrossRefGoogle Scholar
  6. Neumann, W. and Reid, A. (1992a). Arithmetic of hyperbolic manifolds. In Topology ‘80 pages 273–310, Berlin. de Gruyter.Google Scholar
  7. Reid, A. (1990). A note on trace fields of Kleinian groups. Bull. London Math. Soc, 22: 349–352.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Riley, R. (1975). A quadratic parabolic group. Math. Proc. Cambridge Phil. Soc, 77: 281–288.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Riley, R. (1979). An elliptical path from parabolic representations to hyperbolic structures, pages 99–133. Lecture Notes in Mathematics No. 722. Springer-Verlag, Heidelberg.Google Scholar
  10. Riley, R. (1982). Seven excellent knots, pages 81–151. L. M. S. Lecture Note Series Vol. 48. Cambridge University Press, Cambridge.Google Scholar
  11. Thurston, W. (1979). The geometry and topology of three-manifolds. Notes from Princeton University.Google Scholar
  12. Floyd, W. and Hatcher, A. (1982). Incompressible surfaces in punctured torus bundles. Topology and its Applications, 13: 263–282.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Cremona, J. (1984). Hyperbolic tesselations, modular symbols and elliptic curves over complex quadratic fields. Compos. Math, 51: 275–323.MathSciNetzbMATHGoogle Scholar
  14. Burde, G. and Zieschang, H. (1985). Knots Studies in Mathematics Vol. 5. de Gruyter, Berlin.Google Scholar
  15. Rolfsen, D. (1976). Knots and Links. Publish or Perish, Berkeley, CA.Google Scholar
  16. Bowditch, B., Maclachlan, C., and Reid, A. (1995). Arithmetic hyperbolic surface bundles. Math. Annalen, 302: 31–60.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Andreev, E. (1970). On convex polyhedra in Lobachevskii space. Math. USSR, Sbornik, 10: 413–440.CrossRefGoogle Scholar
  18. Hodgson, C. (1992). Deduction of Andreev’s theorem from Rivin’s characterization of convex hyperbolic polyhedra. In Topology ‘80,pages 185–193, Berlin. de Gruyter.Google Scholar
  19. Jones, K. and Reid, A. (1998). Minimal index torsion free subgroups of Kleinian groups. Math. Annalen, 310: 235–250.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Conder, M. and Martin, G. (1993). Cusps, triangle groups and hyperbolic 3-folds. J. Austr. Math. Soc, 55: 149–182.MathSciNetzbMATHCrossRefGoogle Scholar
  21. Chinburg, T. and Reid, A. (1993). Closed hyperbolic 3-manifolds whose closed geodesics all are simple. J. Differ. Geom, 38: 545–558.MathSciNetzbMATHGoogle Scholar
  22. Vinberg, E. (1985). Hyperbolic reflection groups. Russian Math. Surveys, 40: 31–75.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Jorgensen, T. (1977). Compact 3-manifolds of constant negative curvature fibering over the circle. Annals of Math, 106: 61–72.CrossRefGoogle Scholar
  24. Helling, H., Mennicke, J., and Vinberg, E. (1995). On some generalized triangle groups and three-dimensional orbifolds. Trans. Moscow Math. Soc, 56: 1–21.MathSciNetGoogle Scholar
  25. Maclachlan, C. and Reid, A. (1987). Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups. Math. Proc. Cambridge Phil. Soc, 102: 251–257.MathSciNetzbMATHCrossRefGoogle Scholar
  26. Mednykh, A. and Vesnin, A. (1995). Hyperbolic volumes of the Fibonacci manifolds. Siberian Math. J, 2: 235–245.MathSciNetGoogle Scholar
  27. Mednykh, A. and Vesnin, A. (1996). Fibonacci manifolds as two-fold coverings over the three dimensional sphere and the Meyerhoff-Neumann conjecture. Siberian Math. J, 3: 461–467.MathSciNetGoogle Scholar
  28. Thomas, R. (1991). The Fibonacci groups revisited, pages 445–456. L. M. S. Lecture Notes Series Vol. 160. Cambridge University Press, Cambridge.Google Scholar
  29. Fine, B. and Rosenberger, G. (1986). A note on generalized triangle groups. Abh. Math. Sem. Univ. Hamburg, 56: 233–244.MathSciNetzbMATHCrossRefGoogle Scholar
  30. Baumslag, G., Morgan, J., and Shalen, P. (1987). Generalized triangle groups. Math. Proc. Cambridge Phil. Soc, 102: 25–31.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Helling, H., Kim, A., and Mennicke, J. (1998). A geometric study of Fibonacci groups. J. Lie Theory, 8: 1–23.MathSciNetzbMATHGoogle Scholar
  32. Maclachlan, C. and Martin, G. (2001). The non-compact arithmetic triangle groups. Topology, 40: 927–944.MathSciNetzbMATHCrossRefGoogle Scholar
  33. Weeks, J. (1985). Hyperbolic structures on 3-manifolds. PhD thesis, Princeton University.Google Scholar
  34. Matveev, V. and Fomenko, A. (1988). Constant energy surfaces of Hamilton systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds. Russian Math. Surveys, 43: 3–24.MathSciNetzbMATHCrossRefGoogle Scholar
  35. Chinburg, T., Friedman, E., Jones, K., and Reid, A. (2001). The arithmetic hyperbolic 3-manifold of smallest volume. Ann. Scuola Norm. Sup. Pisa, 30: 1–40.MathSciNetzbMATHGoogle Scholar
  36. Chinburg, T. (1987). A small arithmetic hyperbolic 3-manifold. Proc. Am. Math. Soc, 100: 140–144.MathSciNetzbMATHCrossRefGoogle Scholar
  37. Maclachlan, C. and Waterman, P. (1985). Fuchsian groups and algebraic number fields. Trans. Am. Math. Soc, 287: 353–364.MathSciNetzbMATHCrossRefGoogle Scholar
  38. Takeuchi, K. (1971). Fuchsian groups contained in SL(2, Q). J. Math. Soc. Japan, 23: 82–94.MathSciNetCrossRefGoogle Scholar
  39. Takeuchi, K. (1977a). Arithmetic triangle groups. J. Math. Soc. Japan, 29: 91–106.MathSciNetzbMATHCrossRefGoogle Scholar
  40. Baskan, T. and Macbeath, A. (1982). Centralizers of reflections in crystallographic groups. Math. Proc. Cambridge Phil. Soc, 92: 79–91.MathSciNetzbMATHCrossRefGoogle Scholar
  41. Maclachlan, C. and Reid, A. (1991). Parametrizing Fuchsian subgroups of the Bianchi groups. Canadian J. Math, 43: 158–181.MathSciNetzbMATHCrossRefGoogle Scholar
  42. Schmutz Schaller, P. and Wolfart, J. (2000). Semi-arithmetic Fuchsian groups and modular embeddings. J. London Math. Soc, 61: 13–24.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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