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Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups

Published online by Cambridge University Press:  24 October 2008

C. MacLachlan  and
A. W. Reid
C. MacLachlan
Affiliation:
Department of Mathematics, University of Aberdeen, Aberdeen AB9 2TY
A. W. Reid
Affiliation:
Department of Mathematics, University of Aberdeen, Aberdeen AB9 2TY
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 2 , September 1987 , pp. 251 - 257
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Beardon, A. F.. The geometry of discrete groups. Graduate texts in Maths 91 (Springer-Verlag, 1983).CrossRefGoogle Scholar
[2]Borel, A.. Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1981), 1–33.Google Scholar
[3]Brooks, R.. On branched coverings of 3-manifolds which fiber over the circle. J. Reine Angew. Math. 362 (1985), 87–101.Google Scholar
[4]Fine, B.. Fuchsian subgroups of the Picard group. Canad. J. Math. 28 (1976), 481–486.CrossRefGoogle Scholar
[5]Fine, B.. Fuchsian embeddings in Bianchi groups. Preprint.Google Scholar
[6]MacBeath, A. M.. Commensurability of cocompact three-dimensional hyperbolic groups. Duke Math. J. 50 (1983), 1245–1253.CrossRefGoogle Scholar
[7]Maclachlan, C.. Fuchsian subgroups of the groups PSL 2(Od). In Durham Conference on Hyperbolic Geometry, London Math. Soc. Lecture Notes Series no. 112 (Cambridge University Press, 1986), 305–311.Google Scholar
[8]Takeuchi, K.. On some discrete subgroups of SL2(ℝ). J. Fac. Sci. Univ. Tokyo Sec. IA Math. 16 (1969), 97–100.Google Scholar
[9]Takeuchi, K.. A characterisation of arithmetic Fuchsian groups. J. Math. Soc. Japan 27 (1975), 600–612.Google Scholar
[10]Takeuchi, K.. Arithmetic triangle groups. J. Math. Soc. Japan 29 (1977), 91–106.Google Scholar
[11]Takeuchi, K.. Commensurability classes of arithmetic triangle groups. J. Fac. Sci. Univ. Tokyo Sec. IA Math. 24 (1977), 201–212.Google Scholar
[12]Vigneras, M-F.. Arithmétique des algèbres de quaternions. Lecture Notes in Math. Vol. 800 (Springer-Verlag, 1980).CrossRefGoogle Scholar