Professor Michael Kapovich, MSB 2224.

Homepage: http://www.math.ucdavis.edu/~kapovich/RFG/rfg2010.html

CRN for this class is: 30240

Credits: 1 credit for attending and 3 credits for giving a talk.

The goal of this RFG is to be "broad and shallow", covering mostly definitions, concepts, statements

and examples, rather than giving proofs. Almost all talks will be given by students.

The topics will be:

Hyperbolic space: Upper half-space and unit ball models, angles, volume form.

Isometry group of the hyperbolic space. Classification of isometries: Hyperbolic, parabolic, elliptic.

Definition of hyperbolic manifolds. Hyperbolic structures on the pairs of pants. Existence of hyperbolic structures on surfaces.

Riemann surfaces. Uniformization theorem.

Kleinian groups, quotient manifolds and surfaces.

Fundamental domains. Examples of Kleinian groups: Elementary, fuchsian, quasifuchsian, Appolonian packings

Beltrami differentials and quasiconformal maps in 2d.

Extremal maps between Riemann surfaces.

Teichmuller spaces. Teichmuller space of the torus.

Mapping class group and discreteness of its action on the Teichmuller space. Moduli space.

Fenchel-Nielsen coordinates.

Compactification of the moduli space.

Dehn twists. Classification of homeomorphisms of surfaces.

Geodesic laminations. Hausdorff topology on the space of geodesic laminations.

Measured geodesic laminations and train tracks.

Length of a measured lamination and intersection numbers.

Topology of the space of measured laminations.

Measured foliations and their relation to measured laminations. Fat train-tracks.

Harmonic maps from Riemann surfaces into trees.

Dual tree to a measured lamination. Skora's theorem: Converting trees into laminations.

Thurston's compactification of the Teichmuller space.

Earthquakes and Nielsen realization problem/theorem.

Schwarzian differential equation and quadratic differentials.

Complex-projective structures and holonomy.

Relation of complex-projective structures to quadratic differentials and measured laminations.

Ahlfors finiteness theorem.

Geometrically finite and infinite groups.

Pleated surfaces and how to construct them.

Ends of hyperbolic 3-manifolds, notion of tameness.

Tameness theorem. Ending laminations.

Model manifolds.

Ending Lamination Theorem.

Geometric and algebraic convergence.

Geometric structures on manifolds. Holonomy theorem.

3-dimensional geometries.

Geometrization Theorem.

McShane's Identity

Curve complex

Hyperbolicity of curve complex

References:

Wikipedea

M.Kapovich "Hyperbolic manifolds and discrete groups"

C.McMullen "Teichmuller Theory"

J.Cannon et al "Hyperbolic geometry"

M.Bestvina et al "Convexity of hyperbolic length functions"

Y.Minsky "Kleinian groups"

W.Abikoff "Real-analytic theory of Teichmuller space"

Links to videos from MSRI workshops on Kleinian groups and Teichmuller spaces:

General links:

https://secure.msri.org/calendar/workshops/WorkshopInfo/427/show_workshop

https://secure.msri.org/calendar/workshops/WorkshopInfo/428/show_workshop

Kleinian groups and hyperbolic manifolds:

Yair Minsky , Introductory topics in Kleinian groups and hyperbolic 3-manifolds- Part 1

Yair Minsky , Introductory topics in Kleinian groups and hyperbolic 3-manifolds- Part 2

Yair Minsky , Introductory topics in Kleinian groups and hyperbolic 3-manifolds- Part 3

Kenneth Bromberg , Introductory topics in Kleinian groups and hyperbolic 3-manifolds- Part 1

Kenneth Bromberg , Introductory topics in Kleinian groups and hyperbolic 3-manifolds- Part 2

Kenneth Bromberg , Introductory topics in Kleinian groups and hyperbolic 3-manifolds- Part 3

Jeffrey Brock , Introductory topics in Kleinian groups and hyperbolic 3-manifolds- Part 1

Jeffrey Brock , Introductory topics in Kleinian groups and hyperbolic 3-manifolds- Part 2

Jeffrey Brock , Introductory topics in Kleinian groups and hyperbolic 3-manifolds- Part 3

Anna Wienhard , Mostow rigidity- Part 1

Anna Wienhard , Mostow rigidity- Part 2

Teichmuller spaces:

Anna Lenzhen , Teichmuller Theory and its Metrics

Moon Duchin , The Curve Complex and its Relatives

Genevieve Walsh , Thurston's classification of surface automorphisms - Part 1

Genevieve Walsh , Thurston's classification of surface automorphisms - Part 2