Math 595, Real and Complex Hyperbolic Geometry

Fall 2007

Pictures by the Geometry Center and Rich Schwartz respectively.

Course Description

This minicourse will discuss the geometry, topology, and arithmetic of real and complex hyperbolic manifolds. Hyperbolic space, the geometry which has constant negative curvature, is one of the central examples in Riemannian geometry. Complex hyperbolic space is its "complexification", and is a homogeneous geometry of variable negative curvature. In both cases, it is very interesting to study compact manifolds with metrics locally isometric to these hyperbolic spaces. This is equivalent to studying discrete groups of isometries of these geometries, that is, discrete subgroups of the Lie groups SO(1, n) and SU(1, n) which are cocompact. In addition to being among the most concrete locally symmetric spaces, in both cases there are deep and important connections between the study of such manifolds and other areas of mathematics. For instance, W. Thurston and Perelman showed that, in many cases, the study of the topology of 3-manifolds can often be reduced to studying the geometry of real hyperbolic 3-manifolds. Complex hyperbolic geometry has been used by Prasad and Yeung to classify "fake projective planes" in algebraic geometry, and by Rich Schwartz to construct new examples of "spherical CR structures" on 3-manifolds.

In this course, I will introduce real and complex hyperbolic manifolds by focusing on a series of concrete constructions, both arithmetic and geometric, concentrating on the more mysterious case of complex hyperbolic manifolds. Along the way, I will survey a number of related areas, and talk about some of the big open problems about these manifolds, in particular those questions related to the Virtual Haken Conjecture in 3-dimensional topology, i.e. the existence of finite covers with non-trivial first betti number.


My goal is to make this class both interesting and accessible to graduate students in not just geometry/topology, but also in number theory and algebraic geometry. The only prerequisites are familiarity with smooth manifolds, the fundamental group, and covering spaces (e.g. Math 520 and 525), as well as basic undergraduate mathematics such as elementary complex analysis.


Here is a rough outline of the topics I plan to cover, though final course content will be adapted depending on student interests.


Students registered for the course will need to write a short (2-4 page) paper which will be due on Friday, December 7th. This paper is largely free-form, and can be about any subject related to the content of this course. For instance, it could be a brief account of a result not covered in class, a review of the some related results explaining why they are interesting, a detailed work-out of a proof only sketched in class, or careful solutions to problems alluded to in class or taken from one of the texts above.

Further references

In addition to the sources listed above, there is

W. Goldman, Complex Hyperbolic Geometry Oxford University Press, 1999.

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