Syllabus
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)
 Upper half plane
 Poincare disk
 Klein
 Band
 Hemisphere
 Hyperboloid
 The "squares" model (this is a combinatorial model).
 Any simply connected, noncompact, 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,
 Thickthin decomposition of surfaces,
 Collar lemma
 Spaces of hyperbolic structures on surfaces.
 Nielsen expansion and quasigeodesics.
 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 ThreeManifolds
 The Master, in book form, Threedimensional
geometry and topology. Vol. 1, Princeton Mathematical Series, 35,
Princeton University Press,
 Caroline Series, Hyperbolic
Geometry.
