Introductory description

The module is an introduction to the modern theory of hyperbolic geometry with focus on dimension 2.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

To understand the non-Euclidean geometry of hyperbolic space, tesselations and groups of symmetries of hyperbolic space and hyperbolic geometry on surfaces.

Learning outcomes

By the end of the module, students should be able to:

• 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.

Indicative reading list

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.

Subject specific skills

Knowledge of hyperbolic geometry, abstract geometric reasoning, group actions. Ideal for prospective PhD students in geometry, dynamical systems or algebraic geometry.

Transferable skills

Knowledge of hyperbolic geometry, abstract geometric reasoning, group actions. Ideal for prospective PhD students in geometry, dynamical systems or algebraic geometry.