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 exemplifes the motto.

The serious mathematics in this volume is Teichmüller theory, a theory of Riemann surfaces blending analysis, geometry, topology, and algebra. The book is aimed at readers who have completed at least a year of graduate school, in conformity with the advanced level of the material. The writing keeps such readers in mind in many ways, as we'll see.

The book is dedicated in unusually strong language to Fields medalist William Thurston: "Only one dedication is possible for this book. Thanks, Bill, for teaching us all the meaning of geometry." Indeed, Thurston is behind much of the mathematics in the book, as is evident from the forthcoming second volume's title, "Four theorems by William Thurston." But also some of his philosophical views on mathematics, put forth in a foreword of independent interest, are reflected in the book's communicative writing style.

**Teichmüller theory**

In his foreword, Thurston reminds us that two-dimensional geometry is very different from geometry in higher dimensions. In two dimensions, for example, one has a regular k-gon for every k ≥ 3. In contrast, one has only the five Platonic solids in three dimensions and also only finitely many regular polytopes in each higher dimension. Similarly, the planar k-gons for k ≥ 4 are flexible; for example, a square may be deformed to a rhombus. In higher dimensions, regular polytopes are always rigid. Thurston writes that the specialness of two dimensions "is a feature, not a bug."

Teichmüller theory is the ever-expanding manual to this unique feature of mathematics. I'll indicate its broad lines in the next few paragraphs.

The primary focus of Teichmüller theory is Riemann surfaces of finite topological type, thus surfaces with g handles, m punctures, and n holes, for some triple of non-negative integers (g,m,n). The distinction between a puncture and a hole is analytical rather than topological: a puncture can be filled in with a point while a hole can only be filled in with a closed disk. The seven cases with 2g+m+n ≤ 2 are as follows.

(g,m,n) |
Surface(s) |
π_{1} |
dim(T_{g,m,n}) |
dim(Aut_{g,m,n}) |
Cover |

(0,0,0) |
The Riemann sphere **Ĉ** |
{0} |
0 |
6 |
**Ĉ** |

(0,1,0) |
The complex plane **C** |
{0} |
0 |
4 |
**C** |

(0,0,1) |
The unit disk **D** |
{0} |
0 |
3 |
**D** |

(0,2,0) |
The punctured plane **C** -{0} |
**Z** |
0 |
2 |
**C** |

(0,1,1) |
The punctured disk **D** -{0} |
**Z** |
0 |
1 |
**D** |

(0,0,2) |
Annuli A_{s} = {z: s<|z|<1} |
**Z** |
1 |
1 |
**D** |

(1,0,0) |
Tori T_{τ} = **C**/<1,τ> |
**Z**^{2} |
2 |
2 |
**C** |

These are exactly the cases where the surfaces have abelian fundamental group. These are also exactly the cases where the surfaces have a positive-dimensional automorphism group. The interval T_{0,0,2}= (0,1) indexing annuli is the first example of a non-trivial Teichmüller space. The complex upper half plane T_{1,0,0} indexing tori is the second example.

The main case is 2g+m+n ≥ 3 where things behave uniformly. In particular, all surfaces with these numerics have the unit disk as universal cover and so inherit a hyperbolic metric of constant curvature –1. Closed loops on these surfaces come in one of four types, according to how they behave under tightening. Trivial loops can be tightened to any point on the surface. Puncture loops can be made arbitrarily short about a unique puncture. Hole loops can be tightened arbitrarily close to the circumference of a unique hole. Generic loops can be tightened to a unique geodesic. There are respectively 1, m, n, and infinitely many homotopy classes of trivial, puncture, hole, and generic loops. There is a length associated to each homotopy class, zero in the first two cases and positive in the last two cases.

In general, the Teichmüller space T_{g,m,n} is the space of isomorphism classes of "marked" Riemann surfaces of type (g,m,n). Here a marking is certain extra discrete data which gives a label to each homotopy class of closed loops in a natural way. Intuitively, as one moves on a path through T_{g,m,n} the corresponding surface is changing its geometry but maintaining a completely fixed topology. By the theory of quasiconformal mappings, any marked Riemann surface can be canonically continuously deformed into any other marked Riemann surface of the same numerical type. Consequently, each T_{g,m,n} is a contractible metric space with any two points connected by a unique geodesic. One can see in many ways that T_{g,m,n} is simultaneously a real analytic manifold of dimension 6g-6+2m+3n+dim(Aut_{g,m,n}). If n = 0, it also naturally a complex manifold. There is a tremendous interconnected net of yet more natural structures, ranging from alternative metrics, to relations among the various T_{g,m,n} and to compactifications of various sorts.

One way to be more explicit in the main case 2g+m+n ≥ 3 is to use the lengths associated to homotopy classes of loops to get infinitely many functions on T_{g,m,n}. Drawing from these natural functions, one can coordinatize T_{g,m,n} in many ways. An important simple example is T_{0,0,3} = R_{>0}^{3}, via the circumferences of the 3 holes. A more complicated example is T_{g,0,0} = R_{>0}^{3g-3} × R^{3g-3}, via lengths of 3g - 3 appropriate closed geodesics and 3g - 3 associated twist parameters. Other approaches to rendering T_{g,m,n} more explicit highlight other aspects of T_{g,m,n}.

Replacing the marking with orderings of the punctures and holes corresponds to quotienting out by a discrete group, the mapping class group Γ_{g,m,n}. The quotient M_{g,m,n} = T_{g,m,n}/Γ_{g,m,n} is then the space of isomorphism classes of Riemann surfaces of type (g,m,n) with ordered punctures and ordered holes. The only non-trivial instance in the setting 2g+m+n ≤ 2 is the case of tori. Here Γ_{1,0,0} = SL_{2}(Z) and the infinite-degree map from the upper half-plane T_{1,0,0} to the corresponding moduli space M_{1,0,0} = C is given by the classical j-function. In general, the spaces M_{g,m,n} are much less accessible than their Teichmüller covers, since most of the standard functions on T_{g,m,n} are not well-defined on M_{g,m,n}. The M_{g,m,n} are also much more complicated than the T_{g,m,n} both locally and globally, as they have singularities and non-trivial cohomology groups.

However, the spaces M_{g,m,0}, unlike their covers T_{g,m,0}, are naturally complex algebraic varieties, and very important ones at that. These moduli spaces M_{g,m,0} are often studied by purely algebraic methods. Teichmüller theory provides a completely different approach. While Volume 2 of Hubbard's book is mainly about applications to three-manifolds, there could equally well be an independent Volume 3 about applications within algebraic geometry.

**"Whole mind" exposition**

A theme in Thurston's foreword is that the way mathematics resides in our brains is quite different from the way we typically commit it to paper. If we all wrote mathematics in a way which actually took into account how humans understand mathematics, then the task of readers trying to make sense of the literature would be much easier. In his typical way, Thurston advances this point very strongly, writing about "how ineffective and denatured the standard ((definition theorem proof)^{n} remark)^{m} style is for communicating mathematics."

In particular, Thurston is adamant that our understanding of mathematics has an emotional component. He writes, "In mathematics, what is intriguing, puzzling, interesting, surprising, boring, tedious, exciting is crucial," since these attitudes actually shape our cognitive understanding. Even more importantly, at least when the subject is something like Teichmüller theory, our understanding has a very large geometric component. Good mathematical writing should include some direct appeals to our "spatial and visual senses," as well as the usual appeals to the "linguistic, symbol-handling areas" of our brains.

Thurston writes that "John Hubbard approaches mathematics with his whole mind," and indeed he does. Definitions, theorems, proofs, and remarks are embedded in a coherent narrative. Geometry is made visual whenever possible.

The extra narration serves to keep the reader oriented. For example, Hubbard introduces quasiconformal maps on page 111 by explaining that it took him a long time to get used to their paradoxical nature: they are smooth enough for some of calculus to hold but too rough for other parts to hold. Intuitive preliminaries like this one help readers interpret the rigorous material which follows.

The geometrical support comes at all levels. At lower levels, we're taught how to canoe and drive cars in the hyperbolic plane (Section 2.3). We likewise learn that hyperbolic trousers fit us better than Euclidean trousers (Figure 3.5.1). At higher levels, we're taught how to think of Beltrami forms as ellipse fields (Figure 4.8.2) and to literally model a Riemann surface with a quadratic differential using ruled paper and tape (Figures 5.3.3-4). Many figures, such as the cover figure, capture central notions in ways that humans naturally understand them; the corresponding text makes sense only after one has internalized the picture.

Hubbard also keeps the reader in mind in ways besides the two which figure prominently into Thurston's foreword. He aims to be self-contained and appeal to as many readers as possible. Accordingly, there is a two page notation summary, a thirteen page basic glossary, and a fourteen page index. The glossary is particularly handy, with seventy-two definition-based entries, starting with "act freely" and ending with "upper semi-continuous." There is also even a ninety page appendix of background advanced topics which come to the forefront only briefly in the main development. These background advanced topics are indeed best isolated in an appendix; the topics are very wide-ranging, from Dehn twists, to holomorphic functions on Banach manifolds, to Serre duality.

It should be emphasized that keeping the reader in mind does not at all mean skirting difficult points. In fact, Hubbard's aim is to present complete preliminaries and complete proofs, even all the way through the end of the second volume. This self-containment at times becomes quite demanding of the reader. For example, much of the text concerns arbitrary Riemann surfaces, not just those of finite topological type, and this necessarily complicates the presentation.

**Conclusion**

In his preface, Hubbard cites books by Ahlfors (1966), Abikoff (1980), Nag (1988), Imayoshi & Taniguchi (1992), and Gardiner & Lakic (1999). He says they are excellent and recommends them highly. At this point, readers do indeed have many excellent introductions to Teichmüller theory. The volume under review is already a highly competitive newcomer to the list. It will become even more attractive when its sequel volume appears.

David Roberts is an associate professor of mathematics at the University of Minnesota, Morris.