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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 102, Issue 2
  • September 1987, pp. 251-257

Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups

  • C. MacLachlan (a1) and A. W. Reid (a1)
  • DOI:
  • Published online: 24 October 2008

Arithmetic Fuchsian and Kleinian groups can all be obtained from quaternion algebras (see [2,12]). In a series of papers ([8,9,10,11]), Takeuchi investigated and characterized arithmetic Fuchsian groups among all Fuchsian groups of finite covolume, in terms of the traces of the elements in the group. His methods are readily adaptable to Kleinian groups, and we obtain a similar characterization of arithmetic Kleinian groups in §3. Commensurability classes of Kleinian groups of finite co-volume are discussed in [2] and it is shown there that the arithmetic groups can be characterized as those having dense commensurability subgroup. Here the wide commensurability classes of arithmetic Kleinian groups are shown to be approximately in one-to-one correspondence with the isomorphism classes of the corresponding quaternion algebras (Theorem 2) and it easily follows that there are infinitely many wide commensurability classes of cocompact Kleinian groups, and hence of compact hyperbolic 3-manifolds.

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[1]A. F. Beardon . The geometry of discrete groups. Graduate texts in Maths 91 (Springer-Verlag, 1983).

[4]B. Fine . Fuchsian subgroups of the Picard group. Canad. J. Math. 28 (1976), 481486.

[6]A. M. MacBeath . Commensurability of cocompact three-dimensional hyperbolic groups. Duke Math. J. 50 (1983), 12451253.

[9]K. Takeuchi . A characterisation of arithmetic Fuchsian groups. J. Math. Soc. Japan27 (1975), 600612.

[10]K. Takeuchi . Arithmetic triangle groups. J. Math. Soc. Japan29 (1977), 91106.

[12]M-F. Vigneras . Arithmétique des algèbres de quaternions. Lecture Notes in Math. Vol. 800 (Springer-Verlag, 1980).

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