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This volume is the sixth in a series dedicated to Teichmüller theory in a broad sense, including various moduli and deformation spaces, and the study of mapping class groups. It is divided into five parts:
Part A: The metric and the analytic theory.
Part B: The group theory.
Part C: Representation theory and generalized structures.
Part D: The Grothendieck–Teichmüller theory.
Part D: Sources.
The topics surveyed include Grothendieck’s construction of the analytic structure of Teichmüller space, identities on the geodesic length spectrum of hyperbolic surfaces (including Mirzakhani’s application to the computation of Weil–Petersson volumes), moduli spaces of configurations spaces, the Teichmüller tower with the action of the Galois group on dessins d’enfants, and several others actions related to surfaces. The last part contains three papers by Teichmüller, translated into English with mathematical commentaries, and a document that contains H. Grötzsch’s comments on Teichmüller’s famous paper Extremale quasikonforme Abbildungen und quadratische Differentiale.
The handbook is addressed to researchers and to graduate students.