M.Sc Thesis

M.Sc StudentDavidovich Orit
SubjectCongruence Surfaces and the Congruence Subgroup Problem
DepartmentDepartment of Mathematics
Supervisors PROF. Michah Sageev
MR Robert Brooks (Deceased)


In our work we present an alternative approach to the "Congruence Subgroup Problem" - the problem of existence or nonexistence of non-congruence subgroups among those of finite index in.

Our aim is to formulate the problem in geometric terms. Since congruence subgroups provide us with congruence surfaces we ask what geometric properties characterize congruence surfaces among all Riemann surfaces.

The main result of this work is the following. For a Riemann surface S we define a rational invariant by the formula

where denotes the area of S in the hyperbolic metric. We then define an infinite sequence of primes and prove that  for all . Thus every Riemann surface with such a prime value is a possible candidate to a non-congruence surface, and its group of isometries, a possible candidate to a non-congruence subgroup. We then apply our result to construct an infinite family of non-congruence subgroups using the fact that contains a free subgroup of index six.

We note that other geometric characterizations for congruence surfaces do exist. For example, A. Selberg proved that a congruence surfacesatisfies where denotes the first non-zero eigenvalue of the Laplacian.

We connect our result to the problem of a construction of families of compact Riemann surfaces with a large first non-zero eigenvalue of the Laplacian, who are rich enough in the sense that they miss only a finite number of genera. The problem is relevant for the study of the behavior of the first eigenvalue as the genus tends to infinity.