Winter 2015-16: Hyperbolic Geometry

§ Course Overview

This course is an advanced course in Geometry. The aim of the course is to give a proof of Minsky´s Ending Lamination Theorem under the assumption that the injectivity radius is bounded from below. In more elementary terms, we will study hyperbolic 3-manifolds whose fundamental groups are isomorphic to the fundamental group of some hyperbolic surface, and show that there exists an end invariant of such hyperbolic 3-manifolds which completely determines the given manifold up to isometry.

Audience are assumed to know mathematics in the undergraduate level. Abstract algebra (meaning group theory) and the first course in algebraic topology are absolutely necessary, and if you have taken some geometry class before, it would be helpful.

To be a bit more precise, students are assumed to know (at least) what the following words mean:

Riemannian metric, Gaussian curvature, sectional curvature, isometry, geodesic, manifold, (co)tangent bundle, covering space, homotopy/isotopy, fundamental group, homology group, conformal structure/map, homeomorphim/diffeomorphism, etc.

There will be quite a bit of Teichmüller theory going on this course but some of the details will have to be omitted. Fortunately, there will be a lecture course on Teichmüller theory by Dr. Sebastian Hensel in SS 2015-16, so my course should serve as a motivation to learn more about this fantastic subject!

§ Time and location

We meet every Wednesday at Room 1.007 at Mathematisches Institut (Endenicher Allee 60).

The class will start at 14:15 and end at 16:00, and it will have a 15-min break from 15:00 to 15:15 (so it consists of two 45-min lectures).

§ Schedule Issues

- First class is on 28.10.2015 not 21.10.2015 due to Panorama conference.

- There is no class on 02.12.2015 since it is Dies Academicus.

- On 16.12.2015, a guest lecture by Dr. Ilya Gekhtman.

- Christmas break is from December 24, 2015 to January 6, 2016.

- The lecture on 23.12.2015 is replaced by a reading assignment.

- The final exam is on 10.02.2016. The exam location is Room 2.003 at Endenicher Allee 60.

1:30-1:45 D.S. / 1:45-2:00 L.P. / 2:00-2:15 A.B. / 2:15-2:30 M.T. / 2:30-2:45 E.W.

3:00-3:15 E.F. / 3:15-3:30 P.K. / 3:30-3:45 P.S. / 3:45-4:00 B.F.

§ References

There is no textbook for this course. Here are some references which are helpful for background reading.

- M. Kapovich. Hyperbolic manifolds and discrete groups.

- A. Marden. Outer circles.

- J. Ratcliffe. Foundations of hyperbolic manifolds.

- W. Thurston. The geometry and topology of 3-manifolds.

- F. Bonahon. Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots.

- Matsuzaki & Taniguchi. Hyperbolic manifolds and Kleinian groups.

§ Examination

It is likely that we will have a short oral exam at the end of the semester. There will be homework occasionally but it will not be collected.

I will add more information about the course here, so please stay tuned!