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.
Prerequisites
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.
Outline
Here is a rough outline of the topics I plan to cover, though final
course content will be adapted depending on student interests.
Weeks 1-2: Overview of real and complex hyperbolic spaces,
their isometry groups, and basic examples of lattices. For
concreteness, I will focus of real dimensions 2 and 3, and complex
dimensions 1 and 2. Statement of the Thurston/Perelman Geometrization
Theorem. References:
W. Thurston, Chapter 2 of Three-dimensional geometry and
topology, vol 1, Princeton University Press, 1997.
D.B.A. Epstein, Complex Hyperbolic Geometry, in Analytical and
geometric aspects of hyperbolic space (Coventry/Durham, 1984),
93--111, Cambridge Univ. Press, 1987
John Parker, Luminy notes on Complex Hyperbolic Geometry.
Week 3: The arithmetic construction of compact real and
complex hyperbolic manifolds, using Mahler's compactness criterion and
Selberg's Lemma. Brief discussion of the general theory of lattices in
semi-simple Lie groups. Gromov and Piatetski-Shapiro's construction
of non-arithmetic real hyperbolic manifolds in every dimension.
References:
D. Witte Morris, Introduction to Arithmetic Groups. Book draft, 2007.
Weeks 4-5: Thurston gave a very elegant construction of
certain real and complex hyperbolic manifolds that arise as moduli
spaces of polygons and polyhedra, respectively. (In the complex case,
these are the same as those introduced by Picard and Deligne-Mostow.)
I will explain these ideas following:
as well as papers of Yamashita, Nishi, and Kojima which describe the
simpler real hyperbolic case.
Weeks 6-7: From several points of view, it is very useful
to be able to find explicit presentations for the fundamental group of
a real or complex hyperbolic manifolds. In the real hyperbolic case,
this can be done using Dirichlet domains and Poincare's Fundamental
Polyhedron Theorem. The complex hyperbolic case is much more subtle,
and I will discuss some computational methods of Mostow, as refined by
Deraux et. al. in that case.
Week 7: A fundamental conjecture is:
Conjecture: Let M be a compact real or complex
hyperbolic manifold. Then M has a finite cover N where H_{1}(N; Q) is non-zero.
This conjecture is a natural outgrowth of the Virtual Haken
Conjecture in the case of 3-manifolds, which was first proposed by
Waldhausen in the 1960s. This conjecture is particularly mysterious
in the case of complex hyperbolic manifolds. In particular, Mumford
constructed such an arithmetic M, a fake projective
plane, such that all congruence covers N have vanishing
H_{1}(N; Q). This latter remarkable fact
has several proofs both algebro-geometric (Rapoport-Zink) and
based on automorphic forms (Rogawksi/Clozel). Thus the truth of this
conjecture hinges on a negative answer to Serre's Congruence Subgroup
Problem for such M. I will end by trying to explain how one
might investigate these issues computationally if one can find a
concrete presentation for the fundamental group of M, in line
with what has been done in the case of 3-dimensional hyperbolic
3-manifolds:
N. M. Dunfield and W. P. Thurston, The Virtual Haken Conjecture: Experiments and Examples. Geom. Topol. 7 (2003) 399-441.
Grading
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.