Oberwolfach Seminar: On Arithmetically Defined Hyperbolic Manifolds

Date:

November 4th - November 10th, 2007

Organizers:

Jürgen Rohlfs, Eichstätt

Joachim Schwermer, Wien

Ulrich Stuhler, Göttingen

Programme:

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:

• construction of arithmetically defined hyperbolic manifolds
• geometry of hyperbolic manifolds, differential forms and de Rham cohomology
• reduction theory, fundamental domains, covolumes
• construction and relationship between various compactifications
• construction of cycles, intersection numbers of those and corresponding homology classes
• cohomology of arithmetic groups
• arithmetically defined hyperbolic 3 - manifolds

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.

Prerequisites:

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

• full name and address, including e-mail address
• present position, university
• name of supervisor of Ph.D. thesis
• a short summary of previous work and interest

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
.