Penner / Harer Combinatorics of Train Tracks

1. Auflage 2016
ISBN: 978-1-4008-8245-8
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 232 Seiten

Reihe: Annals of Mathematics Studies

Measured geodesic laminations are a natural generalization of simple closed curves in surfaces, and they play a decisive role in various developments in two-and three-dimensional topology, geometry, and dynamical systems. This book presents a self-contained and comprehensive treatment of the rich combinatorial structure of the space of measured geodesic laminations in a fixed surface. Families of measured geodesic laminations are described by specifying a train track in the surface, and the space of measured geodesic laminations is analyzed by studying properties of train tracks in the surface. The material is developed from first principles, the techniques employed are essentially combinatorial, and only a minimal background is required on the part of the reader. Specifically, familiarity with elementary differential topology and hyperbolic geometry is assumed. The first chapter treats the basic theory of train tracks as discovered by W. P. Thurston, including recurrence, transverse recurrence, and the explicit construction of a measured geodesic lamination from a measured train track. The subsequent chapters develop certain material from R. C. Penner's thesis, including a natural equivalence relation on measured train tracks and standard models for the equivalence classes (which are used to analyze the topology and geometry of the space of measured geodesic laminations), a duality between transverse and tangential structures on a train track, and the explicit computation of the action of the mapping class group on the space of measured geodesic laminations in the surface.

Autoren/Hrsg.

Penner, Robert C.

Penner, R. C.

Harer, John L.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• Mathematik | Informatik Mathematik Topologie

Inhaltsverzeichnis

Preface

Acknowledgements

Ch. 1 The Basic Theory 3

1.1 Train Tracks 4

1.2 Multiple Curves and Dehn's Theorem 10

1.3 Recurrence and Transverse Recurrence 18

1.4 Genericity and Transverse Recurrence 39

1.5 Trainpaths and Transverse Recurrence 60

1.6 Laminations 68

1.7 Measured Laminations 82

1.8 Bounded Surfaces and Tracks with Stops 102

Ch. 2 Combinatorial Equivalence 115

2.1 Splitting, Shifting, and Carrying 116

2.2 Equivalence of Birecurrent Train Tracks 124

2.3 Splitting versus Shifting 127

2.4 Equivalence versus Carrying 133

2.5 Splitting and Efficiency 139

2.6 The Standard Models 145

2.7 Existence of the Standard Models 154

2.8 Uniqueness of the Standard Models 160

Ch. 3 The Structure of ML[subscript 0] 173

3.1 The Topology of ML[subscript 0] and PL[subscript 0] 174

3.2 The Symplectic Structure of ML[subscript 0] 182

3.3 Topological Equivalence 188

3.4 Duality and Tangential Coordinates 191

Epilogue 204

Addendum The Action of Mapping Classes on ML[subscript 0] 210

Bibliography 214