M.Sc Thesis | |

M.Sc Student | Davidovich Orit |
---|---|

Subject | Congruence Surfaces and the Congruence Subgroup Problem |

Department | Department 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 surface_{}satisfies _{}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.