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MA448 Content

Content: An introduction to hyperbolic geometry, mainly in dimension two, with emphasis on concrete geometrical examples and how to calculate them. Topics include: basic models of hyperbolic space; linear fractional transformations and isometries; discrete groups of isometries (Fuchsian groups); tesselations; generators, relations and Poincaré's theorem on fundamental polygons; hyperbolic structures on surfaces.

Aims: To introduce the beautiful interplay between geometry, algebra and analysis which is involved in a detailed study of the Poincaré model of two-dimensional hyperbolic geometry.

Objectives: To understand

  • the non-Euclidean geometry of hyperbolic space.
  • tesselations and groups of symmetries of hyperbolic space.
  • hyperbolic geometry on surfaces.

Books:

J.W. Anderson, Hyperbolic geometry, Springer Undergraduate Math. Series.

S. Katok, Fuchsian groups, Chicago University Press.

S. Stahl, The Poincaré half-plane, Jones and Bartlett.

A. Beardon, Geometry of discrete groups, Springer.

J. Lehner, Discontinuous groups and automorphic functions. AMS.

L. Ford, Automorphic functions, Chelsea (out of print but in library).

J. Stillwell, Mathematics and its history, Springer.