Foliations and the geometry of 3manifolds
This book gives an exposition of the socalled "pseudoAnosov" theory
of foliations of 3manifolds, 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.
Download .pdf
Zentralblatt MATH review by Athanase Papadopoulos
MathSciNet Review by Thilo Kuessner
EMS newsletter March 2009 review
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
 PseudoAnosov automorphisms of surfaces
 Geometric structures on general mapping tori
 Peano curves
 Laminations and pinching
 Chapter 2: The topology of S^{1}
 Laminations of S^{1}
 Monotone maps
 Pushout of monotone maps
 Pushforward of laminations
 Leftinvariant orders
 Circular orders
 Homological characterization of circular groups
 Bounded cohomology and MilnorWood
 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 S^{1}
 Chapter 3: Minimal surfaces
 Connections, curvature
 Mean curvature
 Minimal surfaces in R^{3}
 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
 Volumepreserving flows and deadends
 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
 PseudoAnosov flows
 Pushpull
 Productcovered 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 onesided branching
 Long markers
 Complementary polygons
 PseudoAnosov flows
 Chapter 10: Peano curves
 The Hilbert space H^{1/2}
 Universal Teichmüller space
 Spaces of maps
 Constructions and Examples
 Moore's theorem
 Quasigeodesic flows
 Endpoint maps and equivalence relations
 Construction of laminations
 Quasigeodesic pseudoAnosov flows
 PseudoAnosov flows without perfect fits
 Further directions
 References
 Index

