Abstract: Bianchi groups are groups of the form SL(2,R) where R is the
ring of integers of an imaginary quadratic field.
They arise naturally in the study of hyperbolic 3-manifolds and of
certain generalizations of the classical modular
forms (called Bianchi modular forms) for which they assume the role of
the classical modular group SL(2,Z).
In this latter sense, the study of Bianchi groups is fundamental for
developing Langlands' programme for
GL(2) beyond totally real fields.
The overall goal of this talk is to give the audience an overview of
some of the fundamental problems in the arithmetic
aspects of the theory of Bianchi groups. After giving the necessary
background, I will start with a discussion of
the problem of
understanding the behavior
of the dimensions of the cohomology of Bianchi groups and their
Next, I will focus on the
amount of the torsion that one encounters in the cohomology . Finally, I
will discuss the
of these torsion classes.