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This is the fourth volume of the series In the tradition of Thurston whose aim is to provide accessible accounts on the work of William P. Thurston (1946-2012), a work which had a profound influence on the geometry and topology of low-dimensional manifolds, dynamics and the theory of Riemann surfaces. Like its companion volumes of the same series, this volume consists of surveys on topics that have been significantly influenced by Thurston's works and ideas. The topics covered here include surface mapping class groups and their actions on various spaces, the Milnor-Wood inequality, the geometry of the three-dimensional Lie group SOL, Thurston's norm on the second homology group of an oriented 3-manifold, geometry and dynamics on spaces of laminations, arithmetical properties of pseudo-Anosov dilatations, holomorphic dynamics, translation surfaces, uniformization, circle packing, spaces of surfaces of constant curvature with conical singularities and the dynamics of character varieties of surface groups. Some of the chapters stress on the foundational aspects whereas others are concerned with more specialized topics. In most of the chapters, open problems are discussed. The 28 authors of the 18 chapters that make up this book are among the leading experts in the subjects treated. This volume, like the preceding ones in the collection, is addressed at the same time to graduate students and confirmed researchers. The series is intended to serve as an authoritative source on the topics covered.