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 book cover

Teichmüller Theory and Applications to Geometry, Topology, and Dynamics

Volume I: Teichmüller Theory

by John H. Hubbard

with contributions by Adrien Douady, William Dunbar, and Roland Roeder, as well as Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska, and Sudeb Mitra

with forewords by Clifford Earle and William Thurston

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Also see  Volume 2: Surface Homeomorphisms and Rational Functions

From the foreword by William Thurston:

I have long held a great admiration and appreciation for John Hamal Hubbard and his passionate engagement with mathematics....This book develops a rich and interesting, interconnected body of mathematics that is also connected to many outside subjects. I commend it to you....I only wish that I had had access to a source of this caliber much earlier
in my career.

read the entire foreword

From the foreword to volume 1 by Clifford Earle:

...remarkably self-contained... No other book proves both Royden's theorem about automorphisms of Teichmüller spaces and Slodkowski's theorem about holomorphic motions. But the most important novelty is provided by the author's taste for hands-on geometric constructions and the enthusiasm with which he presents them.

see the complete foreword

From the review in Zentralblatt MATH: invaluable  book. It treats  a wonderful subject, and it is written by a great mathematician. It is now an essential reference for every student and every researcher in the field.

see the complete review

From the MAA review:

There is an ambitious new publishing house on the mathematics scene, Matrix Editions, with lead author John H. Hubbard. As a motto, Matrix Editions has chosen "Serious mathematics, written with the reader in mind."  The volume under review is the first volume of a two-volume book.  It beautifully exemplifies the motto.

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From Mathematical Reviews: 

In the volume under review, the author presents Teichmüller theory in order to understand the proofs of Thurston's theorems, but as a result, the book becomes an excellent textbook on Teichmüller theory.

    This volume begins with a foreword by Thurston in which the importance and the beauty of Teichmüller theory are discussed from the point of view of his mathematical experiences and his philosophy on mathematics.  The foreword itself is worth reading....

In conclusion, the reviewer is convinced that this well-written book will be very useful not only for people who are not familiar with Teichmüller theory, but also for the experts.  This is because the reader is offered everywhere in the volume  the deep insights of the author, who looks at the topics developed from a new vantage point.

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From the author's preface:

Between 1970 and 1980, William Thurston astonished the mathematical world by announcing the four theorems discussed in this book:

  • The classification of homeomorphisms of surfaces.
  • The topological characterization of rational maps.
  • The hyperbolization theorem for 3-manifolds that fiber over the circle.
  • The hyperbolization theorem for Haken 3-manifolds.

Not only are the theorems of extraordinary beauty in themselves, but the methods of proof Thurston introduced were so novel and displayed such amazing geometric insight that to this day they have barely entered the accepted methods of mathematicians in the field.


The book is divided into two volumes. The first sets up the Teichmüller theory necessary for discussing Thurston's theorems; the second proves Thurston's theorems, providing more background material where
necessary, in particular for the two hyperbolization theorems.


I have tried very hard to make this book accessible to a second-year graduate student: I am assuming the results of a pretty solid first year of graduate studies, but very little beyond, and I have included appendices with proofs of anything not ordinarily in such courses. I never refer to the
literature for some difficult but important result. Such references are the bane of readers, who often find the sight differences of assumptions and incompatible notations an insurmountable obstacle.

complete preface in html, including list of prerequisites

complete preface in pdf, including list of prerequisites

Front cover (designed by Leila Joiner)

table of contents (html)

table of contents (pdf)

index (in pdf)  

Samples pages from each chapter and the appendices

Errata and notes

Infrequently asked questions (pdf)

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