Rigidity Theory Seminar
We meet 10:30 am-12:30 pm on Tuesdays and Fridays in the conference room in the math graduate offices in Jones.
Here is a bare outline of the first part of the seminar, in which we prove Mostow's rigidity theorem for closed hyperbolic manifolds. I will fill this in with more details later. If you're interested in giving a lecture, please volunteer! The first week is geometric, the second is heavy on analysis, and the third returns to the geometry.
References for part 1
- Lecture 1 (Nick): What is hyperbolic space? The models of hyperbolic space, the classification of hyperbolic isometries, examples of hyperbolic manifolds. (Notes from Lecture 1)
- Lecture 2 (Ian): Beginning the proof of Mostow's Theorem. Outline of the proof. The boundary at infinity. The Morse-Mostow lemma. Construction of the boundary map.
- Lecture 3 (Clark): The theory of quasiconformal mappings in dimension 2 and above. Starting the proof of differentiability.
- Lecture 4 (Clark): Quasiconformal homeomorphisms are absolutely continuous on lines. Sketch of proof of nonsingularity of the Jacobian.
- Lecture 5 (Rachel): Ergodicity of the geodesic flow on closed hyperbolic manifolds. Ergodicity of the boundary action.
- Lecture 6 (Weiyan): Finishing the proof of Mostow Rigidity. If there is time, applications of Mostow's Theorem: Finiteness of isometry group, etc.
- Lecture 7 (Subhadip): The Gromov norm. Straightening of chains. ( Notes from Lecture 7 )
- Lecture 8 (Katie): Gromov's proof of Mostow rigidity
This is a fantastic survey on the subject,
The following will be our primary reading references.
- Notes from Benson Farb's '99 class. I'll see about making these available electronically.
- Thurston's Princeton notes. The relevant chapters for us are 1-3, 5-6.
- Mostow's original paper.
- Lectures on spaces of nonpositive curvature, Werner Ballmann, appendix by M. Brin. The appendix contains a short proof that the geodesic flow on a negatively curved compact manifold is ergodic.
Most of the technical issues in the proof can be trivially bypassed in the case of hyperbolic manifolds, since everything is smooth.
- Lectures on n-dimensional quasi-conformal mappings, Jussi Väisälä. This contains fairly concise proofs of the theorems from analysis that Thurston/Farb put in a black box. The theorems are disguised by the language however; the relevant material is in chapter 4.
- My notes on the regularity of Quasiconformal mappings. Adapted from Jussi Väisälä's text. Should eventually contain proofs of all the regularity results we need.
- Geodesic flow on H^2. Related to Lecture 5. An old homework set of mine in which I computed the geodesic flow on H^2 and the corresponding Anosov splitting.
- Gromov norm notes. These are notes taken by Bena in a class Farb taught on the Gromov norm.
- Adrien Lucker's thesis. A master's thesis that Farb supervised. The sections that I read contained some errors, so I would be careful reading it.
For the second part of the seminar, we will be looking at Margulis' superrigidity theorem for lattices in semisimple Lie groups, as well as Margulis' theorem that lattices in higher rank semisimple Lie groups are arithmetic.
References for part 2
- Lecture 9 (Ian): An introduction to semisimple Lie algebras.
- Lecture 10 (Clark): Symmetric spaces. Geometric motivations to consider lattices in semisimple Lie groups.
If you need any of these books, email me and I can get you a copy.
- Ergodic Theory and Semisimple Groups, Robert Zimmer. This will be our primary reference
- Introduction to Arithmetic Groups, Dave Witte Morris. Supplementary reference which isn't as technical/dry.
- Ergodic Theory, Groups, and Geometry, Dave Witte Morris and Robert Zimmer. Lectures demonstrating how to put superrigidity and its generalizations to work.