Ma 191j, Real and Complex Hyperbolic Geometry

Spring 2004

 Pictures by the Geometry Center and Rich Schwartz respectively.

• Time and Place: Tues-Thur at 10-11:30 am in Sloan 253b (the conference room)
• Instructor: Nathan Dunfield
• E-mail:
• Office: Sloan 258 Office Phone: 4339
• Office Hours: Monday 2-3:30 and by appointment. Or just drop by – even if I can't talk to you then, we'll set up a time to meet.
• Web page: www.its.caltech.edu/~dunfield/classes/2004/191/

Course Description

Hyperbolic space, the geometry which has constant negative curvature, is one of the central examples in Riemannian geometry and has deep connections to the theory of 3-manifolds. Complex hyperbolic space is its "complexification", and is a homogeneous geometry of variable negative curvature. In both cases, it is very interesting to try to understand discrete groups of isometries of these geometries, that is, discrete subgroups of the Lie groups SO(1, n) and SU(1, n). Equivalently, we want to understand manifolds with metrics locally isometric to these hyperbolic spaces. When the quotient has finite volume (i.e. when the group is a lattice in the associated Lie group), they are very rigid; Mostow showed that if two such quotients of dim > 2 have the same fundamental group then they are isometric. On the other hand, when the quotient has infinite volume they frequently are quite flexible. Moreover, in this case, it is interesting to understand the dynamics of the action of the group on the sphere at infinity, contrasting the regions where the action is chaotic with those where the action is discrete. I will try to cover the following topics:
• Weeks 1-2: Overview of real and complex hyperbolic spaces, their isometry groups, and basic examples of lattices. For concreteness, I will focus of real dimensions 2 and 3, and complex dimensions 1 and 2. In the complex case, this will mostly follow:

D.B.A. Epstein, Complex Hyperbolic Geometry, in Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), 93--111, Cambridge Univ. Press, 1987

• Weeks 3-4: Thurston gave a very elegant construction of certain lattices in real and complex hyperbolic space based on looking at configuration spaces of polygons and polyhedra, respectively. (In the complex case, these are the same as the Deligne-Mostow examples.) I will explain these ideas following:

W. P. Thurston, Shapes of polyhedra and triangulations of the sphere, Geometry and Topology Monographs.

as well as papers of Yamashita, Nishi, and Kojima which describe the simpler real hyperbolic case.

• Weeks 4-5: This course will focus more on flexibility than rigidity. I'll start with Thurston's Hyperbolic Dehn Surgery Theorem. This is something special to dimension 3 whereby a non-cocompact finite-volume hyperbolic 3-manifold can be perturbed to an infinite family of compact hyperbolic 3-manifolds. This will follow the appendix to

Boileau and Porti, Geometrization of 3-orbifolds of cyclic type, Asterisque 272.

• Weeks 6-10: The rest of this course will be devoted to understanding Rich Schwartz's work on the interplay between real and complex hyperbolic geometry. In particular, he shows that there are many complex hyperbolic 4-manifolds whose boundary at infinitely is a compact real hyperbolic 3-manifold. This gives examples of compact hyperbolic 3-manifolds which admit a spherical CR-structure, which is a particularly rigid type of contract structure. I will follow his monograph and the condensed version thereof:

R. Schwartz, Spherical CR Geometry and Dehn Surgery, research monograph, 2003, 190 pages.

R. Schwartz, The modular group, the Whitehead link, and spherical CR geometry, 2004, 46 pages.

Prerequisites

Familiarity with smooth manifolds, elementary Riemannian geometry, and a little knowledge of the hyperbolic plane (which you can obtain from reading chapter 2 of Thurston's book listed below).

The course grade will be determined by a final paper. This will be a 6-8 page paper written on a topic in related to the content of this class. The topic will be chosen in consolation with me in the 5th or 6th week of class, and will be due on Friday, June 4. Here are some ideas, with sources:
• Mostow Rigidity (mentioned above): Munkholm, "Gromov's proof of Mostow following Thurston", Springer Lecture Notes in Math, number 788. See aslo Thurston's original notes (available at msri.org) and Ratcliffe's book.
• Deformation and classification of infinite volume hyperbolic 3-manifolds: Yair Minsky, "Combinatorial and Geometrical Aspects of Hyperbolic 3-Manifolds", arXiv.org:math.GT/0205173. A good book on this is Matsuzaki and Taniguchi, "Hyperbolic Manifolds and Kleinian Groups".
• Existence of non-arithmetic lattices for all real hyperbolic spaces. See original paper by Gromov and Piatetski-Shapiro in IHES, 1988.
• The general construction of arithmetic lattices for real hyperbolic 3-space: Maclachlan and Reid, "The Arithmetic of Hyperbolic 3-manifolds", Springer book.
• Construction of hyperbolic structures on 3-manifolds in practice. Weeks, Computation of Hyperbolic Structures in Knot Theory. arXiv.org:math.GT/0309407.
• Aspects of complex hyperbolic space from the point of view of complex and Kahler geometry. E.g. the Kobayashi metric, holomorphic curvature, Kahler hyperbolicity, etc. This links in with Ma 157b.

Scheduling

There will be no class the week of May 3rd. To make up for this, class will continue through June 3rd. Alternatively, extra classes will be scheduled.

Further references

In addition to the sources listed above, there is

W. Goldman, Complex Hyperbolic Geometry Oxford University Press, 1999.

For a good intro to real hyperbolic geometry, see Chapter 2 of

W. Thurston, Three-dimensional geometry and topology, vol 1, Princeton University Press, 1997.