Pseudomodular surfaces

A Fuchsian group is a discrete subgroup of PSL(2,R). As such it acts discontinuously on H (the upper half plane model of the hyperbolic plane) by fractional linear transformations. This action induces an action on the real line. It is well known that if an isometry of H fixes a point of the real line then the point is one of a pair, in the case that the isometry is hyperbolic or the isometry in question is parabolic and the point in question is unique. Points fixed by parabolic elements of a Fuchsian group Γ shall be referred to as the cusps of Γ. If Γ < PSL(2, k) and k is the smallest such field, then consideration of the equation which must be satisfied by a fixed point shows that a cusp must always lie inside k ∪ {∞}. A classical case where the cusp set is completely understood is the case when Γ = PSL(2,Z), and the cusp set coincides with Q ∪ {∞}. More generally determining the cusp set has been hard to do, with only some moderate success—there is a large literature on this type of problem, see for example [10], [11], [15] and [16] to name a few. Recall that Fuchsian or Kleinian groups Γ1 and Γ2 are commensurable if Γ1 has a subgroup of finite index which is conjugate to a subgroup of finite index in Γ2. This paper is motivated by the following question: