Introduction

A Fuchsian group is a discrete subgroup F of PSL2(ℝ) or a conjugate of such a group in PSL2(ℂ). A discrete subgroup G of PSL2(ℂ) is elementary if any two elements of infinite order (regarded as linear fractional transformations) have at least one common fixed point. This is equivalent to the fact that the commutator of any two elements of infinite order has trace 2. The structure of elementary subgroups of PSL2(ℝ) is well-known {see [6]} so for this paper we concentrate on non-elementary groups and use the term Fuchsian group to refer to a non-elementary discrete subgroup F of PSL2(ℝ) or a conjugate of such a group in PSL2(ℂ).

The purpose of this note is to present a complete classification, in one location, of the possibilities for generating pairs for two-generator Fuchsian groups. Specifically we prove the four main theorems listed below. These results have appeared in many different locations {see [26] for a discussion} but it would be convenient and important to have the proofs in just one place. The techniques we employ are straightforward and depend only on the properties of linear fractional transformations and their traces. Before stating the theorems we need some basic ideas from both Fuchsian group theory and abstract combinatorial group theory.