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:

Ergodicity and mixing

Irrational rotations and the ergodic theorem

Entropy and shifts

Geometry of hyperbolic manifolds

The geodesic and horocycle flows on
hyperbolic manifolds

The HoweMoore theorem and the geodesic flow

Ratner's theorem and the horocycle flow

Orbit counting and equidistribution

Amenability, expansion and the Laplacian

The spectrum of the Laplacian,
especially

Amenability

The Selberg trace formula

The problem of isospectral manifolds

Poincaré's operator

Kazhdan's Property T

Expanding graphs

Invariant measures

Hyperbolic manifolds: flexibility and rigidity

PattersonSullivan measure

Hausdorff dimension of limit sets

Convex cocompact groups

Mostow rigidity

Teichmüller theory injects into ergodic
theory

The Ahlfors measure zero conjecture

Sullivan's no invariant line field theorem

Dimension 2 for geometrically infinite groups
(BishopJones)

The ConnesSullivan quantum integral
Books.
Required texts:
[Bus],
[Lub],
[CFS],
[Otal].
Recommended:
[Bea],
[HV],
[Sar].
Very useful but outofprint:
[GGP],
[Nic],
[Zim].
Notes.
Thurston's notes [Th1]
[Th2] may also be of
interest. These notes can be ordered from MSRI
(Nancy Shaw, 15106420143;
nancy@msri.org
).
Additional references.
The following references may be useful:
[BJ],
[Bow],
[Br],
[Ghys],
[Mc],
[Ma],
[Pan],
[Sul].
C. T. McMullen
Sun Jun 23 10:21:08 PDT 1996