# Handbook of Teichmüller Theory: Volume I

Share this page *Edited by *
*Athanase Papadopoulos*

A publication of the European Mathematical Society

The Teichmüller space of a surface was introduced by O.
Teichmüller in the 1930s. It is a basic tool in the study of Riemann's
moduli spaces and the mapping class groups. These objects are fundamental in
several fields of mathematics, including algebraic geometry, number theory,
topology, geometry, and dynamics.

The original setting of Teichmüller theory is complex analysis. The
work of Thurston in the 1970s brought techniques of hyperbolic geometry to the
study of Teichmüller space and its asymptotic geometry.
Teichmüller spaces are also studied from the point of view of the
representation theory of the fundamental group of the surface in a Lie
group \(G\), most notably \(G=\mathrm{PSL}(2,\mathbb{R})\) and
\(G=\mathrm{PSL}(2,\mathbb{C})\). In the 1980s, there evolved an
essentially combinatorial treatment of the Teichmüller and moduli spaces
involving techniques and ideas from high-energy physics, namely from string
theory. The current research interests include the quantization of
Teichmüller space, the Weil-Petersson symplectic and Poisson geometry of
this space as well as gauge-theoretic extensions of these structures. The
quantization theories can lead to new invariants of hyperbolic 3-manifolds.

The purpose of this handbook is to give a panorama of some of the most
important aspects of Teichmüller theory. The handbook should be useful to
specialists in the field, to graduate students, and more generally to
mathematicians who want to learn about the subject. All the chapters are
self-contained and have a pedagogical character. They are written by leading
experts in the subject.

A publication of the European Mathematical Society. Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in analysis.