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


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.


Kleinian Group Fuchsian Group Quaternion Algebra Algebraic Integer Incompressible Surface 
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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
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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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