Oberwolfach Seminar: On Arithmetically Defined Hyperbolic Manifolds

November 4th - November 10th, 2007
Jürgen Rohlfs, Eichstätt
Joachim Schwermer, Wien
Ulrich Stuhler, Göttingen
This seminar will focus on several aspects of the theory of arithmetic groups with an emphasis on the relations among geometric aspects of the corresponding locally symmetric spaces and questions in the arithmetic of algebraic groups. The organizers intend to illustrate the richness of methods and results in this area of research by an investigation of the class of locally symmetric spaces that are quotients of hyperbolic n - space. This includes hyperbolic 3 - manifolds as well as classical arithmetic quotients that arise via lattices coming from certain quadratic forms over algebraic number fields.

An orientable hyperbolic n - manifold is isometric to the quotient of hyperbolic n - space Hn by a discrete torsion free subgroup Γ of the group of orientation - preserving isometries of Hn. The group Γ is said to have finite covolume if Hn/Γ has finite volume, and is said to be cocompact if Hn/Γ is compact.

Among hyperbolic manifolds, the ones originating with arithmetically defined groups Γ form a family of special interest. These arithmetic groups fall naturally into two classes. They can be distinguished by the compactness or non - compactness of the corresponding manifold Hn/Γ. In the latter case the arithmetically defined group Γ has finite covolume.

This subject is related in various ways with number theory, geometry and topology, and, in turn, there are close connections with the theory of automorphic forms. Our objects of concern may be viewed as a most prominent example in the theory of arithmetic groups and their corresponding locally symmetric spaces. Thus, this seminar may very well serve as an introduction to the fruitful interaction between geometric and arithmetic questions, methods and results in the realm of this theory. It is intended to cover the following topic areas where there have been a number of important recent developments:

In addition, there will be a few more advanced lectures on methods developed in the realm of the theory of automorphic forms which might help in understanding the specific case of arithmetically defined hyperbolic manifolds.

A solid knowledge of the theory of differentiable manifolds, the basics in algebraic number theory and some knowledge in Lie theory.
Deadline for applications:
September 30th, 2007

The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 24. The Institute covers accommodation and food. Travel expenses cannot be reimbursed. Applications including

should be sent as hard copy or by e-mail (.ps or .pdf file) to:

Prof. Dr. Gert-Martin Greuel
Universität Kaiserslautern
Fachbereich Mathematik
Erwin Schrödingerstr.
67663 Kaiserslautern, Germany

Mathematisches Forschungsinstitut Oberwolfach   updated: August 22nd, 2007