Hyperbolic Manifolds and Kleinian Groups

Aktuelles

  • Additional Lecture

    The additional lecture this week will take place on Thursday, 12-14 in room B005.
    We’ll finish the proof of Mostow Rigidity for closed 3-manifolds.

    Zuletzt verändert: Mo 01 Aug 2022 12:00

Beschreibung

Hyperbolic manifolds are manifolds carrying a complete Riemannian metric of constant curvature -1 and Kleinian groups are (slight generalisations of) their fundamental groups in dimension 3, seen as groups of isometries of hyperbolic space. In dimension 3, hyperbolic manifolds are both everywhere and interesting: in a precise sense almost every 3-dimensional compact manifold is hyperbolic, and among all 3-manifolds the topology of the hyperbolic ones shows the most complicated behaviour.

In this (2SWS) course, we will develop the theory of Kleinian groups from the ground up. Topics will include
- Examples and constructions of hyperbolic manifolds (e.g. knot complements, gluing of polyhedra, combination theorems)
- Basic structural results for the geometry (thick-thin decomposition, Jørgensen inequality)
- Geometric Finiteness Properties
- The structure of ends
However, the topics can also be adapted to the interest and background of the audience.

This class is aimed at all of the following
- students of Differential Geometry (or Riemannian Geometry) interested in learning about a very important class of negatively curved manifolds, which can be understood in much more detail than the general case
- students of Topology, who are interested in learning about Thurston’s philosophy that “geometry can help in understanding topology”
- students of my 3-manifold class last semester, who want to learn the geometric side of the story

I will assume that participants know:
- differential manifolds (manifolds, tangent and tensor bundles, Riemannian metrics, properly discontinuous group actions)
- covering space theory (existence and classification of covering spaces via the fundamental group)
In particular, participation in my previous 3-manifold course is not a prerequisite.

Institut
Mathematisches Institut
Dozent:in
Kursteilnehmer:innen
15
Anmeldung

Fr 01 Apr 2022 00:00 – Fr 30 Sep 2022 23:59

Abmeldung nur bis Fr 30 Sep 2022 23:59

Material

Aktuell gibt es zu diesem Kurs keine Übungsblätter, oder nur Übungsblätter auf die Sie keinen Zugriff haben (z.B. aufgrund von Fristen bzgl. der Sichtbarkeit).

Material zum Kurs finden Sie hier: Material

Das Kursmaterial ist ohne Anmeldung frei zugänglich