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Special Lecture on Mathematical and Information Sciences I
( MCSHANE GREGORY PATR  )

Mon 5-6Session W832

Outline of lecture

We will develop the theory of surfaces from the point of view of hyperbolic geometry
following the books of Beardon, Buser and Stillwell. After basic material we will discuss lengths
of geodesics, pants decompositions and Fenchel-Nielsen theory. If the time permits we will cover
applications to the study of the group of surface diffeomorphisms and moduli space.

Purpose of lecture

The objective of this course is to give an introduction to
the theory of hyperbolic surfaces and low dimensional topology in general.

Plan of lecture

The course is divided into 4 parts as follows:

Section I. Groups and actions

In this section we will be interested in basic topological concepts
necessary to discuss the results in the second half of the course.
We will follow Beardon's book.

1. Topological groups.
2. Discrete subgroups.
3. The group $PSL(2,\RR)$ as the group of conformal automorphisms of $\HH$.
4. Gluing polygons, Euler characteristic and genus.
5. Riemann's Uniformisation Theorem and the fundamental group of a surface.
6. Fuchsian groups, Schottky groups.
7. Surfaces as quotients and fundamental domains.

Section II. Hyperbolic geometry

This section deals with metric properties of hyperbolic space and how to
do calculations on a hyperbolic surface by "lifting" to the universal cover.

1. The hyperbolic plane, the ideal boundary.
2. Classification of isometries.
3. Hyperbolic trigonometry.
4. Comparison with Euclidean geometry.
5. Closed geodesics on a hyperbolic surface.

Section III. The limit set of a Fuchsian group

One can obtain many interesting results by studying the action of
a fuchsian group on the ideal boundary of the hyperbolic plane.
The smallest closed invariant subset is called the limit set.

1. Classification of points of the limit set.
2. Action a fuchsian group on its limit set (minimality, ergodicity).
3. Measure of the limit set and Basmajian's identities.
4. The space of geodesics of a hyperbolic surface, Louiville measure.

Section IV. The Fricke space of a surface

This is an introduction to the deformation theory of surfaces and their representations.
This is often called teichmueller theory but in the cases we will study it is more correct
to call it Fricke theory. We will follow Goldman's exposition of Fricke's work.

1. The Fricke space of a pair of pants.
2. The Fricke space of a pair of a punctured torus.
3. The action of diffeomorphisms on Fricke space.
4. McShane's identity for a holed torus.

Textbook and reference

A. BEARDON; The geometry of discrete groups. Graduate Texts in Mathematics
P. BUSER; Geometry and spectra of compact Riemann surfaces, Birkhauser
D. MUMFORD, C. SERIES and D. WRIGHT; Indra's Pearls, Cambridge University Press.
J. STILLWELL; Geometry of surfaces, Springer

Related and/or prerequisite courses

will mention at the first lecture on April 9th.

Evaluation

to be announced

Comment from lecturer

More detailed up dated course description can be found at
https://www-fourier.ujf-grenoble.fr/~mcshane/geometry.html