Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-24T13:50:24.994Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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