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Arithmetic Kleinian Groups

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

Abstract

In this chapter, arithmetic Kleinian groups are described in terms of quaternion algebras. An almost identical description leads to arithmetic Fuchsian groups. Both of these are special cases of discrete groups which arise from the group of elements of norm 1 in an order in a quaternion algebra over a number field. Such groups are discrete subgroups of a finite product of locally compact groups, which will be shown, using the results of the preceding chapter, to give quotient spaces of finite volume. Suitable arithmetic restrictions on the quaternion algebras then yield discrete subgroups of SL(2, ℂ) and SL(2, ℝ) of finite covolume and in this way, the existence of arithmetic Kleinian and arithmetic Fuchsian groups is obtained.

Keywords

Number Field Division Algebra Kleinian Group Fuchsian Group Quaternion Algebra 
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

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  18. Gehring, F., Maclachlan, C., Martin, G., and Reid, A. (1997). Arithmet‑ icity, discreteness and volume. Trans. Am. Math. Soc., 349:3611–3643.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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