Topics in Topology: Rigidity theorems.

Some useful references:

• Riemannian geometry, by do Carmo, Birkhauser (1992). The standard first text for learning Riemannian geometry; covers all the Riemannian results we are likely to use in class. Very readable.

• Lectures on hyperbolic geometry, by R. Benedetti and C. Petronio, Springer-Verlag (1992). A great introduction to hyperbolic geometry; covers a lot of material while still staying fairly readable. Includes Gromov's proof of Mostow's rigidity theorem.

• The geometry and topology of 3-manifolds, by W. Thurston, available at the MSRI website. This is fundamental reading material for anyone interested in hyperbolic geometry and/or 3-manifolds. There is a lot of good stuff here, almost all of which is highly relevant to current research.

• Manifolds of nonpositive curvature, by P. Eberlein, U. Hamenstadt, and V. Schroeder, appeared in Proc. Sym. Pure Math. V. 54, Part III, AMS (1993). A fairly thorough survey paper on manifolds of nonpositive curvature. Lots of info (with not so many proofs).

• Boundaries of hyperbolic groups, by N. Benakli and I. Kapovich, in Contem. Math. V. 296, AMS, 2002. A survey paper on boundaries of hyperbolic groups. Well worth reading if you are interested in geometric group theory.

• Minimal entropy and Mostow's rigidity theorems, by G. Besson, G. Courtois, and S. Gallot, in Ergodic Theory Dynam. Systems V. 16 (1996), pgs. 623-649. A survey paper on the BCG technique and its (many) applications, including their simple proof of Mostow rigidity.

• The quasi-isometry classification of lattices in semisimple Lie groups, by B. Farb, in Math. Res. Letters, Vol. 4 (1997), pgs. 705-718. Provides a survey of one of the major results in geometric group theory.

• Rigidity of lattices: an introduction, by M. Gromov and P. Pansu, appeared in "Geometric topology: recent developments (Montecatini Terme, 1990)", Lecture Notes in Math. V. 1504, Springer-Verlag, 1991. Provides a great overview of rigidity results for lattices in semi-simple Lie groups. Also includes a version of the original proof of Mostow rigidity theorem.

• Simplicial volume of closed locally symmetric spaces of non-compact type, by J.-F. Lafont and B. Schmidt, to appear in Acta Math., available on my web page. Uses the BCG technique to resolve Gromov's conjecture on the positivity of the simplicial volume for locally symmetric spaces of non-compact type.

• Surgical methods in rigidity, by F. T. Farrell, Springer-Verlag (1996). A very readable introduction to the Farrell-Jones topological rigidity result and related work.

Outline of topics discussed:

• Day 1: Statement of Mostow rigidity for closed hyperbolic manifolds, applications of Mostow rigidity to (1) non-deformation of hyperbolic structures, (2) uniqueness of embeddings of lattices (up to conjugation), and (3) co-Hopf property for lattices.

• Day 2: Brief review of Riemannian geometry, upper half plane model for the hyperbolic plane, PSL(2,R) acts by isometries, vertical lines are geodesics in the upper half plane model.

• Day 3: Orientation preserving isometries of hyperbolic plane coincides with PSL(2,R), transitivity of the action on the unit tangent bundle, SO(2) (mod +/- identity) as a subgroup.

• Day 4: Hyperbolic plane as the symmetric space PSL(2,R)/SO(2), the disk model and the hyperboloid model for the hyperbolic plane, action of the isometry group on the boundary at infinity, geodesics in the upper half plane model and in the disk model, the lattice PSL(2,Z).

• Day 5: Fundamental domain for the lattice PSL(2,Z), congruence subgroups of PSL(2,Z), proof that most congruence subgroups are torsion-free subgroups in PSL(2,Z), proof that the subgroup PSL(2,Z) has cofinite volume. Application: finite index torsion-free subgroups of PSL(2,Z) are free.

• Day 6: Classification of lattices in 2-dimensional Euclidean space (flat toris) up to scaling and rotations, explicit identification of the moduli space with the quotient of the upper half plane model by the lattice PSL(2,Z), abstract identification of the two via the action of PSL(2,R) on the space of lattices of unit covolume. Identification of the space of unit covolume lattices in n-dimensional Euclidean space (up to scaling and rotations) with the quotient of the symmetric space SL(n,R)/SO(n) by the lattice SL(n,Z).

• Day 7: Pair of pants decomposition of orientable surfaces of genus >1, construction of hyperbolic metrics on higher genus surfaces.

• Day 8: Moduli space of hyperbolic metrics on a fixed surface, higher dimensional hyperbolic space, knot complements, Thurston's hyperbolic Dehn surgery theorem.

• Day 9: Discussion of volumes of hyperbolic manifolds, outline of Besson, Courtois, and Gallot's construction of the isometry in Mostow's rigidity theorem. The isometry is constructed as a composite of three maps, (1) Patterson-Sullivan measures, associating to each point in the space a non-singular probability measure on the boundary at infinity, (2) an induced homeomorphism between the boundaries at infinity, allowing us to push forward the measures from the previous step, and (3) the Barycenter map, allowing us to associate to each of the resulting measures on the boundary at infinity a unique point inside the target space.

• Day 10: We started with Step (2) in the BCG map. Discussed Gromov product on metric spaces, introduced a related product on hyperbolic space, defined sequences that escape to infinity, and introduced the boundary at infinity as the quotient space of all sequences modulo a suitable equivalence relation. Showed that for hyperbolic space, there is a bijective map between this boundary at infinity and the boundary of the disk model.

• Day 11: Defined quasi-isometries between metric spaces. Introduced word metrics on finitely generated groups. Showed that isomorphisms between finitely generated groups induce quasi-isometries (in fact bi-Lipschitz maps) with respect to word metrics. Stated the fundamental theorem of geometric group theory.

• Day 12: Outlined how to complete Step (2) in the definition of the BCG map. Defined length spaces and geodesic spaces. Started the proof of the fundamental theorem of geometric group theory.

• Day 13: Finished proof of the fundamental theorem of geometric group theory. Talked about three major theorems in the field: the characterization of groups of polynomial growth (Gromov), rigidity of lattices in semi-simple Lie groups (various people), and the classification of lattices in semi-simple Lie groups up to quasi-isometry (various people).

• Day 14: Defined delta hyperbolic spaces in terms of the Gromov product. Defined notion of thin geodesic triangle. Proved that delta hyperbolic geodesic spaces are precisely those for which all geodesic triangles are uniformly thin.

• Day 15: For delta hyperbolic geodesic spaces, proof that the Gromov product and the geometrically defined product are at finite distance from each other. Definition of the topology on the boundary at infinity. Reduction of the remaining gaps in establishing Step (2) to a single Key Fact: that in a delta hyperbolic geodesic space, every quasi-geodesic is at a uniformly bounded distance (in terms of the quasi-isometry constants) from a geodesic. Proof of Key Fact postponed till (time permitting) the end of the course.

• Day 16: Outline of the original proof of Mostow rigidity: (a) quasi-isometry induces equivariant quasi-conformal homeomorphisms of the boundary, (b) Rademacher-Stepanov theorem for quasi-conformal maps tells you map on boundary is differentiable almost everywhere, (c) Moore ergodicity theorem shows that derivative maps are conformal almost everywhere, (d) conformal maps on the boundary come from isometries of hyperbolic space. Brief summary of properties that Patterson-Sullivan measures should have (equivariance, no atoms, full support, prescribed Radon-Nikodym derivatives).

• Day 17: Construction of the Patterson-Sullivan measures: exponential volume growth rate for metric balls in hyperbolic space, convergence of the Poincare series associated to a uniform lattice, exponential growth of lattices in hyperbolic space.

• Day 18: Verification of properties of Patterson-Sullivan measures: equivariance under the group action, explicit Radon-Nikodym derivatives (sketched). Modulo the Radon-Nikodym derivatives, sketched the proof that the Patterson-Sullivan measures have no atoms and full support on the boundary at infinity.

• Day 19: More heuristics on why the P.-S. measures have no atoms, discussion of Busemann functions, statement of basic properties of Busemann functions.

• Day 20: Proof that Busemann functions are convex, definition of the barycenter of a measure on the boundary at infinity, proof that measures with no atoms (in particular, P.-S. measures) have well defined barycenters.

• Day 21: Beginning of the proof that the BCG map is an isometry: implicit 1-form describing the BCG map, 2-form derivative of the implicit 1-form, upper bound for the Jacobian of the BCG map in terms of the determinants of two geometrically defined linear operators.

• Day 22: Modulo an optimization problem on symmetric positive matrices of trace 1, established that the Jacobian of the BCG map is bounded above by 1. Proof that the BCG map has Jacobian =1 almost everywhere. Started proof that if Jacobian =1 at a point, then the BCG map is an isometry at that point (i.e. preserves the inner products).

• Day 23: Completed proof that the BCG map is an isometry. Brief discussion of generalization: the entropy rigidity conjecture.

• Day 24: Reviewed the BCG proof of Mostow rigidity for closed hyperbolic manifolds. Introduction to simplicial volume, and discussion of applications (control of degrees of maps, co-Hopf property, non-collapsing, positivity of MinVol and of MinEnt).

• Day 25: Outline of how the BCG technique can be used to establish positivity of the simplicial volume for closed locally symmetric spaces of non-compact type.

• Day 26: Further developments: overview of the Farrell-Jones topological rigidity theorem.

• Day 27: Further developments: overview of Margulis super-rigidity theorem.