Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T04:41:58.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-generator arithmetic Fuchsian groups. II

Published online by Cambridge University Press:  24 October 2008

C. Maclachlan  and
G. Rosenberger
C. Maclachlan
Affiliation:
Department of Mathematical Sciences, Edward Wright Building, University of Aberdeen, Aberdeen AB9 2TY, Scotland
G. Rosenberger
Affiliation:
Fachbereich Mathematik, Universitt Dortmund, PF 500500, 4600 Dortmund 50, Germany
Get access Rights & Permissions [Opens in a new window]

Extract

An arithmetic Fuchsian group is necessarily of finite covolume and so of the first kind. From the structure theorem for finitely generated Fuchsian groups those of the first kind which can be generated by two elements are triangle groups, groups of signature (1;q;0) or (1; ; 1) or groups of signature (0;2,2,2,e;0) where e is odd 6. It is known that there are finitely many conjugacy classes of arithmetic groups with these signatures or indeed with any fixed signature 3, 15. In the case of non-cocompact groups, the arithmetic groups are conjugate to groups commensurable with the classical modular group and are easily determined. For the other groups described above the space of conjugacy classes of all such Fuchsian groups of fixed signature can be described in terms of the traces of a pair of generating elements and their product. In the case of triangle groups this space is a single point. This description has been utilized to determine all classes of triangle groups 13 and groups of signature (1;q;0) which are arithmetic 15. In this paper we determine all classes of groups of signature (0;2,2,2,e;0) with e odd which are arithmetic. The techniques involving traces used so profitably in 15 are not so fruitful in this case. Consequently we have in the main resorted to a quite different method which does not rely on having a precise description of the space of conjugacy classes and hence could be applicable to groups other than those which have rank 2. Extensive use is made of results of Borell on the structure of arithmetic Fuchsian groups which require detailed information on the number fields defining the quaternion algebras. Consequently, the results are given in terms of the quaternion algebra which is determined by its defining field and ramification set, and maximal orders in that quaternion algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 1 , January 1992 , pp. 7 - 24
Copyright
Copyright © Cambridge Philosophical Society 1992

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 Borel, A.. Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), 133.Google Scholar
2Long, R.. Algebraic Number Theory (Marcel Dekker, 1973).Google Scholar
3Maclachlan, C. and Rosenberger, G.. Two-generator arithmetic Fuchsian groups. Math. Proc. Cambridge Philos. Soc. 93 (1983), 383391.CrossRefGoogle Scholar
4Neubser, J.. Private communication, computer printout.Google Scholar
5Odlyzko, A.. Unconditional bounds for discriminants. Preprint (1976).Google Scholar
6Purzitsky, N. and Rosenberger, G.. Two-generator Fuchsian groups of genus one. Math. Z. 128 (1972), 245128;CrossRefGoogle Scholar
correction Math. Z. 132 (1973), 261262.Google Scholar
7Pohst, M. and Graf von Schmettow, J.. Private communication, computer printout.Google Scholar
8Pohst, M., Zassenhaus, H. and Weiler, P.. On effective computation of fundamental units. II. Math. Comp. 38 (1982), 293329.CrossRefGoogle Scholar
9Reid, A.. Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds. Preprint.Google Scholar
10Rosenberger, G. and Kalia, N.. Automorphisms of the Fuchsian groups of type (0;2,2,2,q,0). Comm. Algebra 11 (1978), 11151129.Google Scholar
11Schneider, V.. Die elliptischen Fixpunkte zu Modulgruppen in Quaternionenschiefkrpern. Math. Ann. 217 (1975), 2945.CrossRefGoogle Scholar
12Singerman, D.. Subgroups of Fuchsian groups and finite permutation groups. Bull. London Math. Soc. 2 (1970), 319323.CrossRefGoogle Scholar
13Takeuchi, K.. Arithmetic triangle groups. J. Math. Soc. Japan 29 (1977), 91106.CrossRefGoogle Scholar
14Takeuchi, K.. Commensurability classes of arithmetic triangle groups. J. Fac. Sci. Univ. Tokyo. Sect. IA Math. 24 (1977), 201222.Google Scholar
15Takeuchi, K.. Arithmetic Fuchsian groups with signature (1;e). J. Math. Soc. Japan 35 (1983), 381407.CrossRefGoogle Scholar
16Vigneras, M.-F.. Arithmetigue des Algbres de Quaternions. Lecture Notes in Math. vol. 800 (Springer-Verlag, 1980).CrossRefGoogle Scholar