Foliations and the geometry of 3-manifolds

This book gives an exposition of the so-called "pseudo-Anosov" theory of foliations of 3-manifolds, generalizing Thurston's theory of surface automorphisms.

A central idea is that of a universal circle for taut foliations and other dynamical objects. The idea of a universal circle is due to Thurston, although the development here differs in several technical points from Thurston's approach.

This book was published in May 2007 by Oxford University Press in their Mathematical Monograph series. The .pdf is available for download here with their permission, although I encourage you to buy a physical copy if you find this version useful.

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TABLE OF CONTENTS Preface Chapter 1: Surface bundles Surfaces and mapping class groups Geometric structures on manifolds Automorphisms of tori PSL(2,Z) and Euclidean structures on tori Geometric structures on mapping tori Hyperbolic geometry Geodesic laminations Train tracks Singular foliations Quadratic holomorphic differentials Pseudo-Anosov automorphisms of surfaces Geometric structures on general mapping tori Peano curves Laminations and pinching Chapter 2: The topology of S1 Laminations of S1 Monotone maps Pushout of monotone maps Pushforward of laminations Left-invariant orders Circular orders Homological characterization of circular groups Bounded cohomology and Milnor-Wood Commutators and uniformly perfect groups Rotation numbers and Ghys' theorem Homological characterization of laminations Laminar groups Groups with simple dynamics Convergence groups Examples Analytic quality of groups acting on I and S1 Chapter 3: Minimal surfaces Connections, curvature Mean curvature Minimal surfaces in R3 The second fundamental form Minimal surfaces and harmonic maps Stable and least area surfaces Existence theorems Compactness theorems Monotonicity and barrier surfaces Chapter 4: Taut foliations Definition of foliations Foliated bundles and holonomy Basic constructions and examples Volume-preserving flows and dead-ends Calibrations Novikov's theorem Palmeira's theorem Branching and distortion Anosov flows Foliations of circle bundles Small Seifert fibered spaces Chapter 5: Finite depth foliations Addition of surfaces The Thurston norm on homology Geometric inequalities and fibered faces Sutured manifolds Decomposing sutured manifolds Constructing foliations from sutured hierarchies Corollaries of Gabai's existence theorem Disk decomposition and fibered links Chapter 6: Genuine laminations Abstract laminations Essential laminations Branched surfaces Sink disks and Li's theorem Dynamic branched surfaces Pseudo-Anosov flows Push-pull Product-covered flows Genuine laminations Small volume examples Chapter 7: Universal circles Candel's theorem Circle bundle at infinity Separation constants Markers Leaf pocket theorem Universal circles Leftmost sections Turning corners, and special sections Circular orders Examples Special sections and cores Chapter 8: Constructing transverse laminations Minimal quotients Laminations of the universal circle Branched surfaces and branched laminations Straightening interstitial annuli Genuine laminations and Anosov flows Chapter 9: Slitherings and other foliations Slitherings Eigenlaminations Uniform and nonuniform foliations The product structure on the cylinder at infinity Moduli of quadrilaterals Constructing laminations Foliations with one-sided branching Long markers Complementary polygons Pseudo-Anosov flows Chapter 10: Peano curves The Hilbert space H1/2 Universal Teichmüller space Spaces of maps Constructions and Examples Moore's theorem Quasigeodesic flows Endpoint maps and equivalence relations Construction of laminations Quasigeodesic pseudo-Anosov flows Pseudo-Anosov flows without perfect fits Further directions References Index