Berstein seminar in topology, spring 2020:   Mostow's rigidity theorem

hyperbolic 3-space, from the geometry center

Description.
This course aims to cover several proofs and consequences of what I consider one of the most influential and beautiful theorems of the past century: Mostow's strong rigidity.
One formulation of this theorem is "Suppose you have two compact manifolds of dimension at least 3, each with a metric of constant curvature -1. If their fundamental groups are isomorphic, then the manifolds are isometric." In other words, the fundamental group completely determines the geometry of such a manifold: in principle, one should be able to read off all geometric invariants of M (diameter, volume, length of longest closed geodesic, etc.) from a presentation for its fundamental group (!)
Mostow rigidity also has a purely algebraic statement, in terms of lattices in Lie groups, and many more general formulations than the one given above. It was a precursor and inspiration for many ideas in geometric group theory, homogeneous dynamics, and several areas of geometric topology. We will start with little assumed background (some differential geometry is useful) and aim to understand a few different proofs of Mostow's theorem and their consequences.

As is traditional in the Berstein seminar, those attending the class will give presentations of the readings; in this way we will work through the proofs together.

What we did / what we're doing

1. Introduction
   Leture 1: Motivation - geometric strcutures on manifolds.
   Lecture 2: History - what is a rigidity theorem? Notes
2. Basics of hyperbolic geometry
   References:
   Martelli "An introduction to geometric topology" chapter 2.1-2.3. (probably the best starting point if you are new at this)
   Benedetti and Petronio, chapter A (available free online from Cornell Libaries, but not my favorite exposition)
   Thurston's book "Three dimensional geometry and topology" also has a beautiful, if unconventional introduction.
   Notes from Zhen's lecture (mine)
   Some (old) movies of hyperbolic isometries by C. Goodman-Strauss
3. Hyperbolic structures on surfaces
   Reference: chapter 10.4 of the Primer on Mapping class groups (there are many other good referneces, this is a concise intro)
4. Set-up for the proof: quasi-isometries, quasi-geodesics, the Mostow-Morse lemma.
   These ideas are behind a lot of modern geometric group theory, in particular the study of hyperbolic groups!
   References:
   Section 5.9 of Thurston's notes on 3-manifold topology
   Alternative: Benedetti and Petronio's book (link above), beginning of chapter C.
   I also have some handwritten lecture notes, ask if you want a copy.
   
   This set-up concludes with the proof that a homotopy equivalence between two hyperbolic manifolds induces a homeomorphism of the boundary of hyperbolic space. This is used in all versions of the proof!
5. Thurston's account of Mostow's proof
   The goal here is to show the homeomorphism of the boundaries is a conformal map, hence induced by an isometry of hyperbolic space.
   Suggestion: give the proof using the fact that "geodesic flow is ergodic" as a black box, we can talk about ergodicity after. References: Section 5.9 of Thurston's notes on 3-manifold topology
   Thurston's notes give an outline which leaves some analysis and measure theory results as black boxes.
   Here is a reference that quasiconformal implies a.e. differentiable notes of C. Butler (this fact could also be black boxed)
   Suggested general reading for context: What is... a quasi-conformal mapping? A 2-page Notices article, the title is self-explanatory!
   Lecture notes
6. Ergodicity of geodesic flow
   It is a theorem of Anosov that geodesic flow on a manifold of strict negative curvature is ergodic: any invariant set is null or full measure.
   This fact is easier to prove for hyperbolic manifolds than in general, although the general outline is the same. We'll do the proof for surfaces.
   - Lecture notes . Included at the end is a proof of Birkhoff's theorem (modulo one result) following Milnor's notes.
   
   Other References:
   - (for surfaces) Chapter by A. Manning in Ergodic theory, symbolic dynamics, and hyperbolic spaces (book available in the library)
   - Section 5 of Curt McMullen's notes here (this one is very concise)
   - notes (I have not checked for correctness!) of A. Sanchez
   (further reading: some history and a general proof is given here by Brin)
7. Volume of simplices
   Proof for dimension 3, using the Lobachevsky dilogarithm. Reference: Martelli or Benedetti and Petronio.
   Other References:
   Haagerup and Munkholm's paper showing maximal volume simplices are ideal and regular, in all dimensions >2.
   
8. The Gromov norm on homology
   Mostow rigidity is a statement that algebraic data (fundamental group or homotopy type) determines geometric data (isometry class) for hyperbolic manifolds. Gromov's proof works by defining an algebraic invariant (a pseudo-norm on the homology of manifolds) and showing that, for hyperbolic manifolds, is a genuine norm and the norm of the fundamental class is essentially the volume of the manifold.
   - What is the Gromov (pseudo)-norm?
   - Examples: norm of fundamental class of S^n is zero, same for any manifold with a self-map of degree >1.
   - Gromov's Theorem: For a hyperbolic n-manifold, norm of the fundamental class is volume/v_n, where v_n is the maximal volume of a regular ideal simplex.
   (For n=2, Gromov's theorem gives you Gauss-Bonnet. For n>2, not all ideal simplices have the same norm!)
   References: See below
9. Using Gromov's theorem to prove Mostow rigidity
   References:
   Excellent expository account by Munkholm
   Alternative: Benedetti and Petronio's book, chapter C. This also gives a very detailed explanation, I would suggest looking at both! Martelli also has a chapter that is essentially the same content of Benedetti and Petronio
   Gromov's original work is his paper entitled "Volume and bounded cohomology". It would be nice to cover some selections from this if we are feeling ambitious.
10. (possible topic) Further applications of Gromov norm, bounded cohomology
    Papers chosen depending on interest.
11. Besson-Courtois-Gallot and the barycenter method
    Paper: "Minimal entropy and Mostow's rigidity theorem" (a survey) in Ergodic Theory and Dynamical Systems V. 16 (1996).
    A conceptually simple and beautiful proof of a stronger version of Mostow rigidity. More analytic in flavor, but doesn't use quasi-conformal mappings.
12. (possible topic) Tukia's "zooming in" argument, and some commentary on "Mostow rigidity on the line": what goes right and wrong in 2-dimensional hyperbolic space.
    References:
    Tukia's argument is given in Drutu and Kapovich's geometric group theory book, chapter 22.
    Gehrig and Martin's book "An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings" has a nice self-contained chapter giving a proof of Mostow using this argument
    "Mostow rigidity on the line" is the title of a survey paper of S. Agard, outlining results of Tukia, Sullivan, and others.
13. (possible topic) Semisimple lie groups, algebraic approaches to rigidity theorems.
    Now with added references!
    Introduction: an introduction to symmetric spaces through SL(n,R) Notes by R. Schwartz
    A broader (and quicker) introduction: Chapter 1 of Dave Morris' book on arithmetic groups .
    Main reference: A note on local rigidity by Bergeron and Gelander.
    This is an elegant, simple proof of Calabi-Weil local rigidity using the Ehresmann/Thurston notion of a G,X structure. A great and very readable paper!