MAT 645: Hyperbolic Geometry

Topics

• What does straight mean?
  
• Geometry and geometric structures.
  
• Models (or maps, in the cartography sense) of the hyperbolic plane (or space)
  
1. Upper half plane
2. Poincare disk
3. Klein
4. Band
5. Hemisphere
6. Hyperboloid
7. The "squares" model (this is a combinatorial model).
   
8. Any simply connected, non-compact, Rieman surface which is not the complex plane, has a metric that will make it a model of the hyperbolic plane.
   
• In each of the models,
  
• 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 two sides.
• Inversions are hyperbolic reflections.
• The visual sphere and the sphere at infinity in hyperbolic space
• Mobius transformations, isometries of the disk and the upper half plane
• Convexity,
• Hyperbolic polygons,
• Hyperbolic trigonometry,
• Geometry of surfaces of constant negative curvature,
• Closed geodesics,
• Thick-thin decomposition of surfaces,
• Collar lemma
• 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 groups.
• Cusps, funnels and cone points.
• Mostow rigidity theorem - idea of the proof.

References

• Canon, Kenoyn, Floyd, Hyperbolic Geometry.
• The Master, W. Thurston, The Geometry and Topology of Three-Manifolds
• The Master, in book form, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press,
  
• Caroline Series, Hyperbolic Geometry.