Two generator Fuchsian groups of genus one

Abstract

Let A, BsLf(2,1R), the group of linear fractional transformations with real entries and determinant 1. The results of Knapp [2], Purzitsky [8], and Rosenberger [10] give conditions so that we may determine in all cases but one whether the group {A, B} generated by A and B is a Fuchsian group. The remaining case is when A and B are hyperbolic transformations whose commutator [A, B] is of finite order. We give necessary and sufficient conditions for this group to be discrete. We also obtain all faithful representations of a group whose presentation is {A, B] [A, B]" = 1 } by a discrete subgroup of PSL(2, IR) = SL(2, IR)/{ ___ 1 } ~Lf(2,1R) and partition the representations into disjoint conjugacy classes. Here 1 is the identity. Results in this direction were first given by Lehner and Newman [4, 5] who considered the case of two elliptic generators. Their results were extended to all two generator free products by Purzitsky [-9] and Rosenberger [10]. The form of these distinct conjugacy classes provides an explicit solution to a problem which appears in [1]. Let