MAT 645: Hyperbolic Geometry
- What does straight mean?
- Geometry and geometric structures.
- Models (or maps, in the cartography sense) of the hyperbolic
plane (or space)
In each of the models,
- Upper half plane
- Poincare disk
- The "squares" model (this is a combinatorial model).
- Any simply connected, non-compact, Rieman surface which is not
the complex plane, has a metric that will make it a model of the
Inversions are hyperbolic reflections.
The visual sphere and the sphere at infinity in hyperbolic space
Mobius transformations, isometries of the disk and the upper
Geometry of surfaces of constant negative curvature,
Thick-thin decomposition of surfaces,
Spaces of hyperbolic structures on surfaces.
Nielsen expansion and quasi-geodesics.
Fundamental domains, side pairings, Poincare Theorem.
Discrete subgroups of isometries. Limits sets of discrete
Cusps, funnels and cone points.
Mostow rigidity theorem - idea of the proof.
- Determine which are the straight lines (geodesics)
- Circles, what are they? what is their length and area?
- What is the are of triangles?
- Consider a triangle and one of its sides. Find an upper bound
of the distance between the a point in the chosen side and the other
- Canon, Kenoyn, Floyd, Hyperbolic
- The Master, W. Thurston, The Geometry and Topology
- The Master, in book form, Three-dimensional
geometry and topology. Vol. 1, Princeton Mathematical Series, 35,
Princeton University Press,
- Caroline Series, Hyperbolic