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 Teichmiiller's theory; and second, Nielsen's 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 Teichmiiller's 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 Haken's 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.

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On the geometry and dynamics of diffeomorphisms of surfaces

The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an analogy, in the context of Teichmuller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces. 
In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in the sense of Gromov. In a certain sense this hyperbolicity is an explanation of why the Teichmuller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short. The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmuller space is "relatively hyperbolic" with respect to this family. A similar relative hyperbolicity result is proved for the mapping class group of a surface. 
We also show that the action of pseudo-Anosov mapping classes on the complex is hyperbolic, with a uniform bound on translation distance.

Geometry of the complex of curves I: Hyperbolicity

Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation

The Ricci flow of special geometries Special and limit solutions Short time existence Maximum principles The Ricci flow on surfaces Three-manifolds of positive Ricci curvature Derivative estimates Singularities and the limits of their dilations Type I singularities The Ricci calculus Some results in comparison geometry Bibliography Index.

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The Ricci Flow: An Introduction

This paper continues a geometric study of Harvey's Complex of Curves, whose ultimate goal is to apply the theory of hyperbolic spaces and groups to algorithmic questions for the Mapping Class Group and geometric properties of Kleinian representations. The authors' previous result that the complex is delta-hyperbolic was hard to apply because the complex is not locally finite; in this paper some tools are developed for overcoming this problem, and a combinatorial mechanism introduced which describes sequences of elementary moves in the graph of markings on a surface. These tools are applied to give a family of quasi-geodesic words in the Mapping Class Group, and a linear bound on the shortest word conjugating two conjugate pseudo-Anosov elements. 
A basic tool in the analysis is a family of subsurface projections, which are roughly analogous to closest-point projections to horoballs in classical hyperbolic space. These projections have a strong contraction property which makes it possible to tie together the geometry of the complex and that of the (infinite) subcomplexes that arise as links of vertices. The resulting layered structure of the complex is controlled by means of a combinatorial device called a hierarchy of geodesics, which is the central construction of the paper.

Geometry of the complex of curves II: Hierarchical structure

Quasiconformal mapping Riemann surfaces Quadratic differentials, Part I Quadratic differentials, Part II Teichmuller equivalence The Bers embedding Kobayashi's metric on Teichmuller space Isomorphisms and automorphisms Teichmuller uniqueness The mapping class group Jenkins-Strebel differentials Measured foliations Obstacle problems Asymptotic Teichmuller space Asymptotically extremal maps Universal Teichmuller space Substantial boundary points Earthquake mappings Bibliography Index

Quasiconformal Teichmuller Theory

Results from Riemann Surface Theory and Quasiconformal Mapping Minimal Norm Properties for Holomorphic Quadratic Differentials The Reich-Strebel Inequality for Fuchsian Groups Density Theorems for Quadratic Differentials Teichmuller Theory Teichmuller's Theorem Teichmuller's and Kobayashi's Metrics Discontinuity of the Modular Group Holomorphic Self-Mappings of Teichmuller Space Quadratic Differentials with Closed Trajectories Measured Foliations Index.

Teichmüller Theory and Quadratic Differentials

Assume τ is a point in the Teichmuller space of a Riemann surface which is compact or obtainable from a compact surface by deleting a finite number of punctures. Let be extermal lengths of two transversely realizable measured folitions on the Riemann surface R r corresponding to the point τ. There is a unique Teichmuller line along which the function is minimum. Teichmuller space embeds into projective classes of vectors of square roots of extremal lengths of simple curves on the base surface. The closure of the image of Teichmuller space under this embedding is compact. Moreover, there is a relationship between the boundary of this embedding and the boundary of the extremal length embedding properly contains the Thruston boundary.

Extremal length geometry of teichmüller space

Schiffer’s interior variation and quasiconformal mapping

The objective is to prove two important inequalities of Riemann surface theory and to present them in such a way that one can see their close relationship. The first inequality is the minimal norm property of quadratic differentials first observed by Marden and Strebel [8]. Actually a somewhat different inequality is given. The minimum norm properties have many applications, one of which is a proof of the result of Hubbard and Masur [7] Concerning the existence of a unique holomorphic quadratic differential whose horizontal foliation and vertical measure realize a given measure class of measured foliations on a given Riemann surface. This result is given in section 7. The method of proof is quite different from the one in [7]. The second important inequality is the main inequality of Reich and Strebel. It has central importance in Teichmiiller theory and can be used to derive the infinitesimal form of Teichmiiller's metric [10]. Moreover, this inequality can be used to show that Teichmiiller's metric is the integral of its infinitesimal form [6]. A proof of this result by other methods was given by O'Byrne in [9]. Both of these inequalities turn out to be consequences of certain properties of the trajectory structure of a holomorphic quadratic differential. For the purposes of the "main inequality" we are able to bypass some of the theory of trajectories by means of an averaging device used by Teichmiiller. This averaging device also appears in the proofs of Teichmiiller's theorem given by Bers [4] and Abikoff [1]. But for the minimum norm property in the form given by Marden and Strebel [8], the detailed theory of trajectories seems to be necessary.

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Frederick P. Gardiner

Papers

Quasiconformal Teichmuller Theory

Teichmüller Theory and Quadratic Differentials

Extremal length geometry of teichmüller space

Schiffer’s interior variation and quasiconformal mapping

Measured foliations and the minimal norm property for quadratic differentials