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**Decorated Teichmüller theory.**
*(English)*
Zbl 1243.30003

The QGM Master Class Series. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-075-3/hbk). xvii, 360 p. (2012).

Decorated Teichmüller theory was started by the author in his paper [“The decorated Teichmüller space of punctured surfaces”, Commun. Math. Phys. 113, 299–339 (1987; Zbl 0642.32012)]. Originally, a decorated surface is a complete hyperbolic surface of finite volume with a nonempty set of cusps and with the choice of a closed horocycle around each cusp. Measuring distances to the distinguished horocycles (which can be regarded as a way of “measuring distances to the cusps”) provides a set coordinates for the set of equivalence classes of such surfaces which turned out to be very useful. More generally, a decoration on a hyperbolic surface can be thought of as a set of real parameters (or even other objects) assigned to cusps or to geodesic boundary components. From the set of equivalence classes of decorated surfaces, one builds a space called decorated Teichmüller space, which typically is a fiber bundle over the classical Teichmüller space.

Decorated Teichmüller theory opened a new field of research, with a wide spectrum of applications. The mapping class group acts on decorated Teichmüller space, preserving a cell-decomposition, discovered by Penner using a convex-hull construction in Minkowski space. The cell decomposition also admits a purely combinatorial description using fat graphs. Several important structures on Teichmüller space extend to the decorated version. For instance, the Weil-Petersson Kähler 2-form admits a particularly simple expression, in terms of the so-called \(\lambda\)-length coordinates, which are global affine coordinates for the decorated Teichmüller space (another result of Penner). This has been successfully used in the computation by Penner of Weil-Petersson volumes of moduli spaces. \(\lambda\)-lengths are also useful in infinite dimensions for analyzing homeomorphisms of the circle and the Teichmüller theory of the punctured solenoid.

Besides its major impact on classical Teichmüller theory and on moduli spaces, the decorated theory has applications to the study of Thompson groups, to profinite group theory, to nilpotent group theory and to 3-manifold invariants. All these subjects are considered in the book under review. Decorated Teichmüller theory has also interesting relations with algebraic number theory, harmonic analysis, topological and conformal field theories, tropical geometry and cluster algebras. Furthermore, this theory played a major role in the quantization theory of Teichmüller space; it is heavily used in the work of Kashaev on the subject. We note finally that there are recent applications of decorated Teichmüller theory in computational biology (RNA and proteins) which were developed by Penner and collaborators.

Decorated Teichmüller theory now, after 25 years of development, is quite mature and it was the right time for the author to produce such a book. This book provides a comprehensive and self-contained introduction to the subject and to its major applications. It also contains three appendices on more specialized topics: (A) The geometry of Gauss products, (B) Dual to the Kähler two-form, and (C) Stable curves and screens.

Several monographs on Teichmüller theory appeared in print in the last three decades, but most of them (if not all) deal with the analytic aspects of the theory. This book is unique because it concentrates on the combinatorial-topological aspect, an aspect which is now as much important as the analytical one.

The book is based on lectures that the author gave at the Center for the Topology and Quantization of Moduli Spaces. The fact that these lectures appear now in print will be of great help for all students and researchers working in the field, and the whole mathematical community will benefit from it. It is very carefully written, and it is now an important part of our mathematical literature.

Decorated Teichmüller theory opened a new field of research, with a wide spectrum of applications. The mapping class group acts on decorated Teichmüller space, preserving a cell-decomposition, discovered by Penner using a convex-hull construction in Minkowski space. The cell decomposition also admits a purely combinatorial description using fat graphs. Several important structures on Teichmüller space extend to the decorated version. For instance, the Weil-Petersson Kähler 2-form admits a particularly simple expression, in terms of the so-called \(\lambda\)-length coordinates, which are global affine coordinates for the decorated Teichmüller space (another result of Penner). This has been successfully used in the computation by Penner of Weil-Petersson volumes of moduli spaces. \(\lambda\)-lengths are also useful in infinite dimensions for analyzing homeomorphisms of the circle and the Teichmüller theory of the punctured solenoid.

Besides its major impact on classical Teichmüller theory and on moduli spaces, the decorated theory has applications to the study of Thompson groups, to profinite group theory, to nilpotent group theory and to 3-manifold invariants. All these subjects are considered in the book under review. Decorated Teichmüller theory has also interesting relations with algebraic number theory, harmonic analysis, topological and conformal field theories, tropical geometry and cluster algebras. Furthermore, this theory played a major role in the quantization theory of Teichmüller space; it is heavily used in the work of Kashaev on the subject. We note finally that there are recent applications of decorated Teichmüller theory in computational biology (RNA and proteins) which were developed by Penner and collaborators.

Decorated Teichmüller theory now, after 25 years of development, is quite mature and it was the right time for the author to produce such a book. This book provides a comprehensive and self-contained introduction to the subject and to its major applications. It also contains three appendices on more specialized topics: (A) The geometry of Gauss products, (B) Dual to the Kähler two-form, and (C) Stable curves and screens.

Several monographs on Teichmüller theory appeared in print in the last three decades, but most of them (if not all) deal with the analytic aspects of the theory. This book is unique because it concentrates on the combinatorial-topological aspect, an aspect which is now as much important as the analytical one.

The book is based on lectures that the author gave at the Center for the Topology and Quantization of Moduli Spaces. The fact that these lectures appear now in print will be of great help for all students and researchers working in the field, and the whole mathematical community will benefit from it. It is very carefully written, and it is now an important part of our mathematical literature.

Reviewer: Athanase Papadopoulos (Strasbourg)

### MSC:

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

30F60 | Teichmüller theory for Riemann surfaces |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

30F10 | Compact Riemann surfaces and uniformization |