Deformations of Fuchsian groups Fall 2000, SUNY Stony Brook Christopher Bishop

1. Introduction

2. Kleinian groups (a) Definition

(b) Limit set

(c) Quotient manifold

(d) Convex core, geometric finiteness

(e) Conical and escaping limit sets

(f) Critical exponent

(g) Patterson-Elstrodt-Sullivan formula

(h) Decay of heat kernel

(i) Ahlfors conjecture

3. Fuchsian groups (a) Classification of elements

(b) Uniformization theorem

(c) First and second kind

(d) Divergence and convergence type

(e) Harmonic functions

(f) Extremal length

4. Hausdorff dimension (a) Definition and examples

(b) Minkowski dimension

(c) Besicovitch-Taylor theorem

(d) Frostman measures

(e) The β’s

(f) Theorem: Uniformly wiggly implies dim > 1.

5. Theorem: escaping rays have dimension 1 (thick case).

6. Theorem: δ = dim(Λ_{c}).

7. Theorem: δ is lower semi-continuous

8. Conformal maps (a) Riemann mapping theorem

(b) Caratheodory’s theorem

(c) Bloch functions

(d) McMillians theorem

(e) Makarov’s theorems

(f) Schwarzian derivatives

(g) The factorization theorem 9. Quasiconformal maps

(a) Definitions

(b) Quasisymmetry

(c) Quasi-invariance of extremal length

(d) The Beltrami equation

(e) The tangent space

(f) Carleson condition implies absolute continuity (g) Mappings of Y -pieces

(h) The dimension of Teichmuller space

(i) Convergence of μ implies uniform convergence (j) Astala’s theorem

(k) Douady-Earle extension theorem

(l) Rough isometries

(m) Sullivan’s convex hull theorem

10. Bowen’s theorem (a) Bowen’s proof

(b) Wiggly set proof

(c) Estimating derivative proof

(d) Divergence type proof

11. Astala-Zinsmeister’s rectifiable examples (a) There is a big deformation of compact support (b) Semmes’ theorem

12. The dimension 1 examples (a) Shrinking a geodesic

(b) Effect on limit set

(c) Limit set has almost no tangents

(d) Limit set has dimension 1.

13. Ruelle’s theorem

14. Astala-Zinsmeister’s convergence group examples 15. Divergence type examples