R. C. Penner, John L. Harer

Combinatorics of Train Tracks. (AM-125), Volume 125

• Broschiertes Buch

Produktbeschreibung

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.

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Produktdetails

• Verlag: Princeton University Press
• Seitenzahl: 232
• Erscheinungstermin: 23. Dezember 1991
• Englisch
• Abmessung: 234mm x 156mm x 13mm
• Gewicht: 360g
• ISBN-13: 9780691025315
• ISBN-10: 0691025312
• Artikelnr.: 21107408

Autorenporträt

R. C. Penner & John L. Harer

Inhaltsangabe

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

Rezensionen

The book is beautifully written, with a clear path of theoretical development amid a wealth of detail for the technician. . . . [T]his text provides a valuable reference work as well as a readable introduction for the student or newcomer to the area.