# Lectures on Representations of Surface Groups

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*François Labourie*

A publication of the European Mathematical Society

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 & Endorsements

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.

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