 
Summary: March 8, 2011 Lecture Note Series, IMS, NUS  Review Vol. 9in x 6in hypstr
HYPERBOLIC STRUCTURES ON SURFACES
Javier Aramayona
School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, Galway. IRELAND
Email: Javier.Aramayona@nuigalway.ie
We give a brief introduction to hyperbolic structures on surfaces. Using
the concepts of developing map and holonomy, we sketch a proof that
every surface equipped with a complete hyperbolic surface is isometric
to a quotient of H by a Fuchsian group. We then define Teichm¨uller
spaces and explain FenchelNielsen coordinates. Finally, we introduce
mapping class groups and show that they act properly discontinuously
on Teichm¨uller space.
1. Introduction
This paper is intended as a brief introduction to hyperbolic structures on
surfaces, Teichm¨uller spaces and mapping class groups. It is based on the
first half of the course "Hyperbolic structures on surfaces", given by C.
Leininger and the author during the programme "Geometry, Topology and
Dynamics of Character Varieties" at the Institute for Mathematical Sci
ences of Singapore in July 2010. It accompanies the article [23], also in this
