# Decorated Teichmüller Theory

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*Robert C. Penner*

A publication of the European Mathematical Society

There is an essentially “tinker-toy” model of a trivial
bundle over the classical Teichmüller space of a punctured
surface, called the decorated Teichmüller space, where the fiber
over a point is the space of all tuples of horocycles, one about each
puncture. This model leads to an extension of the classical mapping
class groups called the Ptolemy groupoids and to certain matrix models
solving related enumerative problems, each of which has proved useful
both in mathematics and in theoretical physics. These spaces enjoy
several related parametrizations leading to a rich and intricate
algebro-geometric structure tied to the already elaborate
combinatorial structure of the tinker-toy model. Indeed, the natural
coordinates give the prototypical examples not only of cluster
algebras but also of tropicalization.

This interplay of combinatorics and coordinates admits further
manifestations, for example, in a Lie theory for homeomorphisms of the
circle, in the geometry underlying the Gauss product, in profinite and
pronilpotent geometry, in the combinatorics underlying conformal and
topological quantum field theories, and in the geometry and
combinatorics of macromolecules.

This volume gives the story a wider context of these decorated
Teichmüller spaces as developed by the author over the last two
decades in a series of papers, some of them in
collaboration. Sometimes correcting errors or typos, sometimes
simplifying proofs, and sometimes articulating more general
formulations than the original research papers, this volume is self
contained and requires little formal background. Based on a master's
course at Aarhus University, it gives the first treatment of these
works in monographic form.

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 the study of geometrical aspects and mathematical foundations of quantum field theory and string theory.