Two-generator arithmetic Fuchsian groups. II

Abstract

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.