Editor's note: this review covers both volume I and volume II of the Handbook of Teichmüller Theory.
I have always thought of Teichmüller theory as one of those new life-forms that sprang from the ever so fertile imagination of Alexander Grothendieck in the years of his self-imposed exile first from the world of mainstream mathematics and eventually from the world itself. This naïve prejudgment on mine testifies not only to my outsider status but also to my unabashed fascination with all things Grothendieck. After all, is there any one on the mathematical scene about whom a more wonderfully bizarre legend has evolved, so much of it already in embryo when he was still a very visible public figure? The lore around Grothendieck and his “methods,” to borrow a phrase from Sherlock Holmes, i.e. his unique and all but magical mathematical style, is irresistible, and is fostered by his sporadic correspondence from his hermitage in the Pyrenees.
I recently had occasion to review Ronald Brown’s Topology and Groupoids in this venue and discovered that the author is a Grothendieck correspondent himself; to be sure, Grothendieck’s influence is explicitly discernible in his book, and, indeed, in the entire subject in question. Which brings us to Grothendieck’s modus operandi since his departure from the academy: always a prolific letter-writer (cf. The Grothendieck-Serre Correspondence, for example), he has not departed from mathematics at all, just from the world of mathematicians, and from time to time he presents some of his marvelous ideas to mathematically sympathetic scholars who are then in a position to spread the word.
And so it has come to pass that one of these scholars is Leila Schneps, who now counts among her specialties Grothendieck’s theory of dessins d’enfants (kids’ drawings, as it were), and these objects are unquestionably part of hypermodern Teichmüller theory. I first came across Grothendieck’s explorations of Teichmüller theory browsing through the pages of Geometric Galois Actions I, II, edited by Schneps and her husband, Pierre Lochak, in particular in Grothendieck’s famous latter, Esquisse d’un Programme, presented in these volumes. Such beautiful ideas, and so much to learn: it was clear then that I had a lot of work ahead of me, and it is lamentably still that way (unfortunately, I haven’t yet had occasion to wed action to thought as regards doing work in this area). In any case, this is why, as I indicated above, I am still an outsider, and this in turn accounts for my surprise that dessins d’enfants don’t occur in the book(s) under review until p. 255 of Part One — p. 255!
What comes before it, then?! Well, we get an idea of the (very) big picture, i.e. what’s really going on, by looking more carefully at p. 255: “…we hope to bring out a general perspective which places dessins within the tautological geometric framework underpinning the patchwork collection of Teichmüller spaces which cover the various modular families… one needs the entire collection of lattices (finite co-volume Fuchsian groups) and Teichmüller space inclusions to reflect the wide range of possible Belyi representations of complex curves defined over [the algebraic closure] of Q: every meromorphic function which has 3 ramification values determines a covering of the complex projective line with the requisite properties and each admissible cusp form of weight 4 produces a Teichmüller disc and, thereby, candidate members of the set of… points of the space of moduli [definable over the rationals’ algebraic closure]. This space may provide an appropriate starting point for the (still incomplete) process of constructing the ultimate arithmetic modular family descrtibed in Grothendieck’s Esquisse…” And then the (algebraic?) geometric fun begins with the section, “Grothendieck dessins and Thurston’s examples,” the author of this chapter being William J. Harvey.
Well, we find out a lot from this excerpt already: Teichmüller spaces have to do with moduli (here, in point of fact, is the first line from the Foreword: “In a broad sense, the subject of Teichmüller theory is the study of moduli spaces for geometric structures on surfaces…”), there’s a classification scheme (Hah!) in the game, Fuchsian groups and therefore automorphic functions get to play, classical and modern function theory is everywhere, and the flavor is, at least in places, algebraic geometric. Can it be otherwise, with Grothendieck on the playing-field?
Back to p. 255: “A dessin d’enfant or simply dessin is defined to be a connected (finite) graph in a compact surface, whose complement is a union of cells and which has a bipartite structure on the vertices… we restrict attention to… a special kind of dessin, concentrating on the subclass of dessins which arise from pulling back the standard triangulation of CP1 through a Belyi function, [i.e.] a holomorphic branched covering mapping… with three critical values, 0, 1, and ∞. This consists of a topological decomposition of the Riemann sphere into two triangles… with vertices given by the above three symbols [encoding 0, 1, ∞]…” — and the tone is set. This is stuff that hearkens back to Riemann, at least philosophically (and was he not the first to work with moduli in the modern sense?), and sits at the intersection of everything from arithmetic geometry and geometric topology to representation theory and… well, more about this presently.
The point to be taken, however, is that dessin d’enfants are only part of the story, of course, and the larger picture of Teichmüller theory manifestly possesses a host of aspects and facets beyond the algebraic geometric connections. In fact, the editor of the books under review, Handbook of Teichmüller Theory I, II, Athanase Papadopoulos, states in his Foreword that the prevailing goal is “to give a comprehensive picture of the classical and of the recent developments in Teichmüller theory.” There’s a pedigree to be taken note of, and a correspondingly broad context. Says Papadopoulos on page v: “This subject makes important connections between several areas in mathematics, including low-dimensional topology, hyperbolic geometry, dynamical systems theory, differential geometry, algebraic topology, representations of discrete groups in Lie groups, symplectic geometry, topological quantum field theory, string theory, and there are others.” Wow!
He continues: “As is well known, [the Teichmüller space of a surface] can be seen from different points of view[:] … equivalence classes of conformal structures on the surface, … equivalence classes of hyperbolic metrics on this surface, … equivalence classes of representations of the fundamental group of the surface into a Lie group … Teichmüller space inherits from these points of view various structures, including several interesting metrics [including] Teichmüller, Weil-Petersson, Thurston, Bergman, Carathéodory, Kähler-Einstein, McMullen, etc. … [as well as] a natural complex structure, a symplectic structure, a real analytic structure, the structure of an algebraic set, cellular structures, various boundary structures, a natural discrete action by the mapping class group, a quantization theory of its Poisson and symplectic structures, a measure-preserving geodesic flow, a horocyclic flow, and the list of structures goes on and on …” From my erstwhile parochial point of view, having previously only glimpsed a little about dessins d’enfants from leafing through the aforementioned book by Schnaps and Lochak, I’m amazed at the span of this subject. This is truly a vast edifice.
More from Papadopoulos, now from the first entry in the Handbook, namely the “Introduction to Teichmüller theory, old and new”: “During a remarkably brief period of time (1935–1941) Teichmüller wrote about thirty papers which laid the foundation of the theory which now bears his name. After Teichmüller’s death in 1943 (at the age of 30), L. V. Ahlfors, L. Bers, H. E. Rauch and several of their students and collaborators started a project that provided a solid grounding for Teichmüller’s ideas. The realization of this project took more than two decades, during which the whole complex-analytic theory … was built. In the 1970s, W. P. Thurston opened a new and wide area of research by introducing beautiful techniques of hyperbolic geometry in the study of Teichmüller space and of its asymptotic geometry … in the 1980s there also evolved an essentially combinatorial treatment of Teichmüller and moduli spaces, involving techniques from … string theory … [C]urrent research interests … include the quantization of Teichmüller space using the Weil-Petersson symplectic and Poisson geometry of this space, as well as gauge-theoretic extensions of these structures …” Again: Wow!
It follows that these Handbooks are anything but light reading, even as they deal with a plethora of different aspects of a gorgeous part of mathematics. Papadopoulos split the collection of articles in these 794 + 874 pages into four parts, namely, the metric and analytic theory, the group theory, and the quantum theory (all to be followed by the phrase “of Teichmüller spaces”), and the theory of surfaces with singularities and discrete Riemann surfaces. These four components are not in themselves simply connected: they are the four headings of parts A–D of Part One, but the first two themes (or is it three? Papadopous says three, but the quantum theoretical aspects seem to disappear spontaneously in the jump between to the second part: a small matter, though) surface again in Part Two.
The level of scholarship in Handbook of Teichmüller Theory, I, II, is obviously uniformly high, adding up to seventeen articles (in depth surveys and then some: lots of proofs) in Part one and nineteen in Part Two, and Papadopoulos’ editing is superb. His introduction alone suffices not only to enlighten any interested mathematician about the sweep of Teichmüler theory and its current developments, but to whet the reader’s appetite dramatically for what lies ahead. I, for one, have already given in completely: I was tempted years back with the Schneps-Lochak book, coming at it because of my recent studies in algebraic geometry; the present work completely reels me in, and I am keen to learn many things I hadn’t imagined touching before, beyond even the titanic reach of post-Grothendieck algebraic geometry. To my mind this material is so pretty that it’s just plain wrong (though admittedly possible, of course) not to go through the whole book from soup to nuts. But even choosing articles selectively will prove to be a rewarding experience: as I mentioned already, the scholarship is of a high quality, and the writing is also very good.
Handbook of Teichmüller Theory, I, II can’t help but become a definitive work of reference in the field.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.