Hyperbolic manifolds, discrete groups and ergodic theory
Math 277 - Berkeley - Fall 1996
C. McMullen
Tenative Course Description
April 1996

Web page: http://math.berkeley.edu/~ctm/math277.

Prerequisites: Intended for advanced graduate students. Acquaintence with hyperbolic geometry, Lie groups, representation theory and functional analysis will be useful.

Course description. We will discuss discrete groups and ergodic theory from many points of view, especially in relation to hyperbolic manifolds of dimensions 2 and 3. Some possible topics include:

1. Ergodicity and mixing
   1. Irrational rotations and the ergodic theorem
   2. Entropy and shifts
   3. Geometry of hyperbolic manifolds
   4. The geodesic and horocycle flows on hyperbolic manifolds
   5. The Howe-Moore theorem and the geodesic flow
   6. Ratner's theorem and the horocycle flow
   7. Orbit counting and equidistribution

2. Amenability, expansion and the Laplacian
   1. The spectrum of the Laplacian, especially
   2. Amenability
   3. The Selberg trace formula
   4. The problem of isospectral manifolds
   5. Poincaré's -operator
   6. Kazhdan's Property T
   7. Expanding graphs
   8. Invariant measures
3. Hyperbolic manifolds: flexibility and rigidity
   1. Patterson-Sullivan measure
   2. Hausdorff dimension of limit sets
   3. Convex cocompact groups
   4. Mostow rigidity
   5. Teichmüller theory injects into ergodic theory
   6. The Ahlfors measure zero conjecture
   7. Sullivan's no invariant line field theorem
   8. Dimension 2 for geometrically infinite groups (Bishop-Jones)
   9. The Connes-Sullivan quantum integral

Books. Required texts: [Bus], [Lub], [CFS], [Otal]. Recommended: [Bea], [HV], [Sar]. Very useful but out-of-print: [GGP], [Nic], [Zim].

Notes. Thurston's notes [Th1] [Th2] may also be of interest. These notes can be ordered from MSRI (Nancy Shaw, 1-510-642-0143; nancy@msri.org).

Additional references. The following references may be useful: [BJ], [Bow], [Br], [Ghys], [Mc], [Ma], [Pan], [Sul].

• References