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Lectures on Representations of Surface Groups
François Labourie, Université Paris Sud, Orsay, France
A publication of the European Mathematical Society.
 Zurich Lectures in Advanced Mathematics 2013; 146 pp; softcover Volume: 17 ISBN-13: 978-3-03719-127-9 List Price: US$38 Member Price: US$30.40 Order Code: EMSZLEC/17 The subject of these notes is the character variety of representations of a surface group in a Lie group. The author emphasizes the various points of view (combinatorial, differential, and algebraic) and is interested in the description of its smooth points, symplectic structure, volume and connected components. He also shows how a three manifold bounded by the surface leaves a trace in this character variety. These notes were originally designed for students with only elementary knowledge of differential geometry and topology. In the first chapters, the author does not focus on the details of the differential geometric constructions and refers to classical textbooks, while in the more advanced chapters proofs occasionally are provided only for special cases where they convey the flavor of the general arguments. These notes might also be used by researchers entering this fast expanding field as motivation for further studies. The concluding paragraph of every chapter provides suggestions for further research. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in surface groups. Reviews "The European Mathematical Society has been publishing compact books like this one for a number of years now, and it is indeed a great service to all mathematicians. The books (at least the ones I've reviewed) are of a high quality and are eminently readable, modulo the right preparation. This book is no exception: it's very well-written and the topics covered are wonderful and deep. Furthermore, Labourie takes a fascinating approach to all this very sexy differential geometry by working in the graph theoretic and combinatorial angles, as indicated. It is an excellent book." -- Michael Berg, MAA Reviews Table of Contents Introduction Surfaces Vector bundles and connections Twisted cohomology Moduli spaces Symplectic structure $$3$$-manifolds and integrality questions Bibliography Index