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On the structure of certain arithmetic subgroups of SL2()

Published online by Cambridge University Press:  24 October 2008

C. Maclachlan
C. Maclachlan
Affiliation:
Department of Mathematics, University of Aberdeen
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Extract

In [6], p. 80, Serre showed that the group SL2(Z[1/p]) is a free product of two copies of SL2(Z) amalgamated along a sub-group of index p + 1 in both components. This result was obtained from the structure theory of groups acting on trees and the construction of a tree on which SL2 over a field with discrete valuation acts in a natural way. This result can be extended to other SL2 examples. Using the above description, Serre [6] also gave presentations for the groups SL2(Z[1/p]) where p = 2, 3.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 2 , March 1985 , pp. 211 - 217
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Helling, H.. Konstruktion arithmetischer Grenzkreisgruppen von erster Art durch die Einheiten der rationalen Quaternionenalgebren. Doctoral Dissertation, Universitat Miinster, 1964.Google Scholar
[2]Helling, H.. On the commensurability class of the rational modular group. J. London Math. Soc. 2 (1970), 67–72.CrossRefGoogle Scholar
[3]Knapp, A. W.. Doubly-generated Fuchsian groups. Michigan Math. J. 15 (1968), 289–304.CrossRefGoogle Scholar
[4]Maclachlan, C. and Rosenberger, G.. Two-generator arithmetic Fuchsian groups. Math. Proc. Cambridge Philos. Soc. 93 (1983), 383–391.CrossRefGoogle Scholar
[5]Purzitsky, N. and Rosenberger, G.. Two-generator Fuchsian groups of genus one. Math. Z. 128 (1972), 245–251.CrossRefGoogle Scholar
[6]Serre, J.-P.. Trees. (Springer-Verlag, 1980.)CrossRefGoogle Scholar
[7]Takeuchi, K.. On some discrete subgroups of SL2(ℝ). J. Fac. Sci. Univ. Tokyo Sect. I. 16 (1967), 97–100.Google Scholar
[8]Takeuchi, K.. A characterisation of arithmetic Fuchsian groups. J. Math. Soc. Japan 27 (1975), 600–612.CrossRefGoogle Scholar
[9]Takeuchi, K.. Arithmetic triangle groups. J. Math. Soc. Japan 29 (1977), 91–106.CrossRefGoogle Scholar
[10]Takeuchi, K.. Arithmetic Fuchsian groups with signature (1; e). J. Math. Soc. Japan 35 (1983), 381–407.CrossRefGoogle Scholar
[11]Vignebas, M.-F.. Arithmétique des algébres de quaternions. Lecture Notes in Math. No. 800 (Springer-Verlag, 1980).CrossRefGoogle Scholar