William P. Thurston

Three dimensional manifolds, Kleinian groups and hyperbolic geometry

1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the

On the geometry and dynamics of diffeomorphisms of surfaces

This article was widely circulated as a preprint, about 12 years ago. At that time the Bulletin did not accept research announcements, and after a couple of attempts to publish it, I gave up, and the preprint did not find a home. I very soon saw that there were many ramifications of this theory, and I talked extensively about it in a number of places. One year I devoted my graduate course to this theory, and notes of Bill Floyd and Michael Handel from that course were circulated for a while. The participants in a seminar at Orsay in 1976-1977 went over this material, and wrote a volume [FLP] including some original material as well. Another good general reference, from a somewhat different point of view, is a set of notes of lectures by A. Casson, taken by S. Bleiler [CasBlei]. There are by now several alternative ways to develop the classification of diffeomorphisms of surfaces described here. At the time I originally discovered the classification of diffeomorphism of surfaces, I was unfamiliar with two bodies of mathematics which were quite relevant: first, Riemann surfaces, quasiconformal maps and Teichmiillers theory; and second, Nielsens theory of the dynamical behavior of surface at infinity, and his near-understanding of geodesic laminations. After hearing about the classification of surface automorphisms from the point of view of the space of measured foliations, Lipman Bers [Bersl] developed a proof of the classification of surface automorphisms from the point of view of Teichmüller theory, generalizing Teichmiillers theorem by allowing the Riemann surface to vary as well as the map. Dennis Sullivan first told me of some neglected articles by Nielsen which might be relevant. This point of view has been discussed by R. Miller, J. Gilman, M. Handel and me. The analogous theory, of measured laminations and 2-dimensional train tracks in three dimensions, has been considerable development. This has been applied to reinterpret some of Hakens work, to classify incompressible surfaces in particular classes of 3-manifolds in papers by me, Hatcher, Floyd, Oertel and others in various combinations. Shalen, Morgan, Culler and others have developed the related theory of groups acting on trees, and its relation to measured laminations, to define and analyze compactifications of representation spaces of groups in SL(2, C) and SO(n, 1); this has many interesting applications, including the theory of incompressible surfaces in 3-manifolds.

On proof and progress in mathematics

Author(s): Thurston, William P. | Abstract: In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.

A presentation for the mapping class group of a closed orientable surface

THE CENTRAL objects of study in this paper are collections {C,, . . . , C,} of g disjoint circles on a closed orientable surface M of genus g, whose complement M-(C, u . . . U C,) is a 2g-punctured sphere. We call an isotopy class of such collections a cut system. Of course, any two cut systems are related by a diffeomorphism of M, by the classification of surfaces. We show that any two cut systems are also joined by a finite sequence of simple moues, in which just one Cj changes at a time, to a circle intersecting it transversely in one point and disjoint from the other Ci’s. Furthermore, we find a short list of relations between sequences of simple moves, sufficient to pass between any two sequences of simple moves joining the same pair of cut systems. From these properties of cut systems it is a routine matter to read off a finite presentation for the mapping class group of M, the group of isotopy classes of orientation preserving self-diffeomorphisms of M. Unfortunately, the presentation so obtained is rather complicated, and stands in need of considerable simplification before much light will be shed on the structure of the mapping class group. Qualitatively, one can at least deduce from the presentation that all relations follow from relations supported in certain subsurfaces of M, finite in number, of genus at most two. This may be compared with the result of Dehn [3] and Lickorish [4] that the mapping class group is generated by diffeomorphisms supported in finitely many annuli. A finite presentation in genus two was obtained by Birman-Hilden[2], completing a program begun by Bergau-Mennicke [l]. For higher genus the existence of finite presentations was shown by McCool[ lo], using more algebraic techniques. For another approach to finite presentations, see [12], and for general background on mapping class groups, see [ 111. Our methods apply also to maximal systems of disjoint, non-contractible, nonisotopic circles on M. This is discussed briefly in an appendix.

Conway's tiling groups

John Conway discovered a technique using infinite, finitely presented groups that in a number of interesting cases resolves the question of whether a region in the plane can be tessellated by given tiles. The idea is that the tiles can be interpreted as describing relators in a group, in such a way that the plane region can be tiled, only if the group element which describes the boundary of the region is the trivial element 1. A convenient way to describe the construction is by means of the Cayley graph or graph of a group. If G is a group, then its graph F(G) with respect to generators g1, g2 . . ., gn is a directed graph whose vertices are the elements of the group. For each vertex v E F(G), there will be n outgoing edges, labeled by the generators, and n incoming edges: the edge labeled gi connects v to vgi. It is convenient to make a slight modification of this picture when a generator gi has order 2. In that case, instead of drawing an arrow from v to vgi and another arrow from vgi back to v, we draw a single undirected edge labeled gi. Thus, in a drawing of the graph of a group, if there are any undirected edges, it is understood that the corresponding generator has order 2. The graph of a group is automatically homogeneous: for every element g E G, the transformation v -4 gv is an automorphism of the graph. Every automorphism of the labeled graph has this form. This property characterizes graphs of groups: a graph whose edges are labeled by a finite set F such that there is exactly one incoming and one outgoing edge with each label at each vertex is the graph of a group if and only if it admits an automorphism taking any vertex to any other. Whenever R is a relator for the group, that is, a word in the generators which represents 1, then if you start from v EF rand trace out R, you get back to v again. If G has presentation

Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds

This is the first in a series of papers showing that Haken manifolds have hyperbolic structures; this first was published, the second two have existed only in preprint form, and later preprints were never completed. This eprint is only an approximation to the published version, which is the definitive form for part I, and is provided for convenience only. All references and quotations should be taken from the published version, since the theorem numbering is different and not all corrections have been incorporated into the present version. Parts II and III will be made available as eprints shortly.

On the existence of contact forms

Using an old theorem of Alexander, we give a short and elementary proof that every closed, orientable 3-manifold has a contact form. Introduction. Let Mm (m = 2n + 1) be a closed, smooth, orientable mmanifold; a contact form on M is a smooth 1-form co such that wo A(dcj)n g 0 at each point. One knows [21 that the existence of a contact form on Mm allows a reduction of the structure group of the tangent bundle of M to U(n) and hence, in particular, the odd-dimensional Stiefel-Whitney classes of M have to vanish [4]. This, however, gives no information in dimension 3 and in [31 Chern 1 asked: Does every closed, smooth, orientable 3-manifold admit a smooth contact form? This question was first answered (in the affirmative) by Lutz [6] and Martinet [7]. They based their proof on the theorem that every closed, orientable 3-manifold can be obtained from S3 by surgery, the same theorem used by Lickorish, Novikov and Zieschang to prove that every such manifold admits a codimension 1 foliation, i.e. a nonzero 1-form 1/ on M such that 71 A d71 0; that is, in dimension 3, the exact opposite of a contact form. It was observed by one of us (see also [5]) that the existence theorem for codimension 1 foliations on M is an immediate consequence of the fol-

Separators for sphere-packings and nearest neighbor graphs

A collection of <italic>n</italic> balls in <italic>d</italic> dimensions forms a <italic>k</italic>-ply system if no point in the space is covered by more than <italic>k</italic> balls. We show that for every <italic>k</italic>-ply system Γ, there is a sphere <italic>S</italic> that intersects at most <italic>O</italic>(<italic>k</italic><supscrpt>1/<italic>d</italic></supscrpt><italic>n</italic><supscrpt>1−1/<italic>d</italic></supscrpt>) balls of Γ and divides the remainder of Γ into two parts: those in the interior and those in the exterior of the sphere <italic>S</italic>, respectively, so that the larger part contains at most (1−1/(<italic>d</italic>+2))<italic>n</italic> balls. This bound of (<italic>O</italic>(<italic>k</italic><supscrpt>1/<italic>d</italic></supscrpt><italic>n</italic><supscrpt>1−1/<italic>d</italic></supscrpt>) is the best possible in both <italic>n</italic> and <italic>k</italic>. We also present a simple randomized algorithm to find such a sphere in <italic>O(n)</italic> time. Our result implies that every <italic>k</italic>-nearest neighbor graphs of <italic>n</italic> points in <italic>d</italic> dimensions has a separator of size <italic>O</italic>(<italic>k</italic><supscrpt>1/<italic>d</italic></supscrpt><italic>n</italic><supscrpt>1−1/<italic>d</italic></supscrpt>). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.

Shapes of polyhedra and triangulations of the sphere

The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a complex hyperbolic cone-manifold. In some interesting special cases, the metric completion is an orbifold. The concrete description of these spaces of shapes gives information about the combinatorial classification of triangulations of the sphere with no more than 6 triangles at a vertex.

Pinching constants for hyperbolic manifolds

SummaryWe show in this paper that for everyn≧4 there exists a closedn-dimensional manifoldV which carries a Riemannian metric with negative sectional curvatureK but which admits no metric with constant curvatureK≡−1. We also estimate the (pinching) constantsH for which our manifoldsV admit metrics with −1≧K≧−H.