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.
KeywordsKleinian Group Fuchsian Group Quaternion Algebra Algebraic Integer Parabolic Element
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- Hejhal, D. (1976). The Selberg Trace Formula for PSL(2,R). Volume 1. Lecture Notes in Mathematics No. 548. Springer-Verlag, Berlin.Google Scholar
- Neumann, W. and Reid, A. (1992a). Arithmetic of hyperbolic manifolds. In Topology ‘80 pages 273–310, Berlin. de Gruyter.Google Scholar
- 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
- Riley, R. (1982). Seven excellent knots, pages 81–151. L. M. S. Lecture Note Series Vol. 48. Cambridge University Press, Cambridge.Google Scholar
- Thurston, W. (1979). The geometry and topology of three-manifolds. Notes from Princeton University.Google Scholar
- Burde, G. and Zieschang, H. (1985). Knots Studies in Mathematics Vol. 5. de Gruyter, Berlin.Google Scholar
- Rolfsen, D. (1976). Knots and Links. Publish or Perish, Berkeley, CA.Google Scholar
- 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
- Thomas, R. (1991). The Fibonacci groups revisited, pages 445–456. L. M. S. Lecture Notes Series Vol. 160. Cambridge University Press, Cambridge.Google Scholar
- Weeks, J. (1985). Hyperbolic structures on 3-manifolds. PhD thesis, Princeton University.Google Scholar