# Handbook of Teichmüller Theory: Volume III

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*Athanase Papadopoulos*

A publication of the European Mathematical Society

The subject of this handbook is Teichmüller theory in a
wide sense, namely the theory of geometric structures on surfaces and
their moduli spaces. This includes the study of vector bundles on
these moduli spaces, the study of mapping class groups, the relation
with \(3\)-manifolds, the relation with symmetric spaces and
arithmetic groups, the representation theory of fundamental groups,
and applications to physics. Thus the handbook is a place where
several fields of mathematics interact: Riemann surfaces, hyperbolic
geometry, partial differential equations, several complex variables,
algebraic geometry, algebraic topology, combinatorial topology,
low-dimensional topology, theoretical physics, and others. This
confluence of ideas toward a unique subject is a manifestation of the
unity and harmony of mathematics.

This volume contains surveys on the fundamental theory as well as
surveys on applications to and relations with the fields mentioned
above. It is written by leading experts in these fields. Some of the
surveys contain classical material, while others present the latest
developments of the theory as well as open problems.

This volume is divided into the following four sections:

- The metric and the analytic theory
- The group theory
- The algebraic topology of mapping class groups and moduli spaces
- Teichmüller theory and mathematical physics

This handbook is addressed to graduate students and researchers in all the fields mentioned.

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

#### Readership

Graduate students and research mathematicians interested in metric and analytic theory, group theory, algebraic topology of mapping class groups and moduli spaces, Teichmüller theory, and mathematicial physics.