Introduction

A Fuchsian group is a discrete subgroup of PSL2(R) and two such groups Γ1,Γ2 are commensurable if and only if the intersection Γ1 ∩ Γ2 is of finite index in both Γ1 and Γ2. In this paper we determine when two two-generator Fuchsian groups of finite covolume are commensurable and in addition, the relationship between two such groups, by obtaining part of the lattice of the commensurability class which contains one representative from each conjugacy class, in PGL2(R), of two-generator groups. By the result of Margulis (see e.g. [Z]) that a non-arithmetic commensurability class contains a unique maximal member, the non-arithmetic cases reduce to a compilation of earlier results on determining which two-generator Fuchsian groups occur as subgroups of finite index in other two-generator Fuchsian groups [Sc], [S2], [R]. For the arithmetic cases, all two-generator arithmetic Fuchsian groups have been determined [T2], [T4], [MR] and one can immediately read off when two such groups are commensurable from the structure of the corresponding quaternion algebra (see e.g. [T3]). The relationship between such groups is more difficult to determine and we utilise structure theorems for arithmetic Fuchsian groups [B], [V]. The relationship between arithmetic triangle groups was determined in [T3].