MAT 626, Topics in complex analysis: deformations of Fuchsian groups

Christopher Bishop

Professor, Mathematics
SUNY Stony Brook

Office: 4-112 Mathematics Building
Phone: (516)-632-8289
Dept. Phone: (516)-632-8290
FAX: (516)-632-7631

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Time and place: MF 10:30-11:50, ESS 177

A Fuchsian group is a discrete group of Mobius transformations acting on the unit disk and a deformation of such a group is a conformal map of the disk onto a region which conjugates the group to another group of Mobius transformations acting on the image. We will be concerned with the properties of such deformations, particularly the geometry of the boundary (i.e., the limit set).

We will start by discussing some relevant background material. Among the topics expect to discuss are: Riemann surfaces and the uniformization theorem, quasiconformal maps and the Beltrami equation, Schwarzian derivatives, boundary behavior of conformal maps, Hausdorff dimension, basic properties of discrete groups, the Poincare exponent, Sullivan's convex hull theorem,...

We are mainly interested in geometric properties of the limit sets of quasiFuchsian groups. There are two main topics which will motivate us. First is Bowen's theorem: this says that if G is a cocompact group then any deformation has a limit set which is either a circle or has Hausdorff dimension strictly greater than one. One of our main goals is to describe several proofs of this and in fact to charaterize all groups for which this property holds.

The second topic is Ruelle's theorem. It says that if G is cocompact then the dimension of the limit set is a (real) analytic function of the deformation. Depending on time, I will try to prove this and will give examples of groups for which it fails. However, it is still an open problem to characterize which groups Ruelle's theorem holds for, and I will discuss the current state of affairs and related open problems.

very rough and preliminary outline of topics we will cover

list of references related to Kleinian groups

glossary of terms related to Kleinian groups

Some preprints of mine which discuss these questions in greater detail include (these are postscript versions):

Hausdorff dimension and Kleinian groups

Divergence groups have Bowen's property

A criterion for the failure of Ruelle's property

Quasiconformal lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture

Compact deformations of Fuchsian groups (with Peter Jones)

Nonrectifiable limit sets of dimension one

Surfaces which approximate the thrice punctured sphere