 
Summary: Pseudomodular surfaces
D. D. Long
and A. W. Reid
September 5, 2001
1 Introduction.
A Fuchsian group is a discrete subgroup of PSL(2, R). As such it acts discontinuously on H2
(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 H2
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 shall be referred to as the cusps of . If < 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 = 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 successthere 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 1 and 2 are commensurable if 1 has a subgroup of
finite index which is conjugate to a subgroup of finite index in 2. This paper is motivated by the
