• 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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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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