For several decades, Teichmüller theory has been one of the most active research areas in mathematics, with a very wide range of points of view, including Riemann surface theory, hyperbolic geometry, low-dimensional topology, several complex variables, algebraic geometry, arithmetic, partial differential equations, dynamical systems, representation theory, symplectic geometry, geometric group theory, and mathematical physics.

This book is the fourth volume in a Handbook of Teichmüller Theory project that started as an attempt to present, in a most comprehensive and systematic way, the various aspects of this theory with its relations to all the fields mentioned. The handbook is addressed to researchers as well as graduate students.

This volume is divided into five parts:

• Part A: The metric and the analytic theory
• Part B: Representation theory and generalized structures
• Part C: Dynamics
• Part D: The quantum theory
• Part E: Sources

Parts A, B, and D are sequels to parts on the same theme in previous volumes. Part E contains the translation together with a commentary of an important paper by Teichmüller that is almost unknown, even to specialists. Making the original ideas of and motivations for a theory clear is crucial for many reasons, and making this translation, together with the commentary that follows, available will give readers a broader perspective on Teichmüller theory.

The various volumes in this collection are written by experts who have a broad view on the subject. In general, the chapters are expository, while some of them contain new and important results.