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 space and of the mapping class group. 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 in the study of Teichmüller space and of 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 = PSL(2,ℝ) and G = PSL(2,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.