Handbook of Teichmüller Theory, Volume IV [electronic resource] / Athanase Papadopoulos

Introduction to Teichmüller theory, old and new, IV / Athanase Papadopoulos -- Local and global aspects of Weil–Petersson geometry / Sumio Yamada -- Simple closed geodesics and the study of Teichmüller spaces / Hugo Parlier -- Curve complexes versus Tits buildings: structures and applications / Lizhen Ji -- Extremal length geometry / Hideki Miyachi -- Compactifications of Teichmüller spaces / Ken’ichi Ohshika -- Arc geometry and algebra: foliations, moduli spaces, string topology and field theory / Ralph M. Kaufmann -- The horoboundary and isometry group of Thurston’s Lipschitz metric / Cormac Walsh -- The horofunction compactification of the Teichmüller metric / Lixin Liu, Weixu Su -- Lipschitz algebras and compactifications of Teichmüller space / Hideki Miyachi -- On the geodesic geometry of infinite-dimensional Teichmüller spaces / Zhong Li -- Holomorphic families of Riemann surfaces and monodromy / Hiroshige Shiga -- The deformation of flat affine structures on the two-torus / Oliver Baues -- Higher Teichmüller spaces: from SL(2,$\mathbb{R}$) to other Lie groups / Marc Burger, Alessandra Iozzi, Anna Wienhard -- The theory of quasiconformal mappings in higher dimensions, I / Gaven J. Martin -- Infinite-dimensional Teichmüller spaces and modular groups / Katsuhiko Matsuzaki -- Teichmüller spaces and holomorphic dynamics / Xavier Buff, Guizhen Cui, Lei Tan -- A survey of quantum Teichmüller space and Kashaev algebra / Ren Guo -- Variable Riemann surfaces / Oswald Teichmüller -- A commentary on Teichmüller’s paper Veränderliche Riemannsche Flächen / Annette A’Campo-Neuen, Norbert A’Campo, Lizhen Ji, Athanase Papadopoulos.

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Teichmüller theory is, since several decades, 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. The present 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. The present 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 of parts on the same theme in previous volumes. Part E has a new character in the series; it contains the translation together with a commentary of an important paper by Teichmüller which is almost unknown even to specialists. Making clear the original ideas of and motivations for a theory is crucial for many reasons, and rendering available this translation together with the commentary that follows will give a new impulse and will contribute in putting the theory into a broader perspective. The various volumes in this collection are written by experts who have a broad view on the subject. In general, the chapters have an expository character, which is the original purpose of this handbook, while some of them contain new and important results.