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Combinatorics of Train Tracks. (AM-125), Volume 125 : Annals of Mathematics Studies (Paperback) - R. C. Penner

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

Annals of Mathematics Studies (Paperback)


Published: 23rd December 1991
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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.

"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."--Zentralblatt f?r Mathematik

The Basic Theoryp. 3
Train Tracksp. 4
Multiple Curves and Dehn's Theoremp. 10
Recurrence and Transverse Recurrencep. 18
Genericity and Transverse Recurrencep. 39
Trainpaths and Transverse Recurrencep. 60
Laminationsp. 68
Measured Laminationsp. 82
Bounded Surfaces and Tracks with Stopsp. 102
Combinatorial Equivalencep. 115
Splitting, Shifting, and Carryingp. 116
Equivalence of Birecurrent Train Tracksp. 124
Splitting versus Shiftingp. 127
Equivalence versus Carryingp. 133
Splitting and Efficiencyp. 139
The Standard Modelsp. 145
Existence of the Standard Modelsp. 154
Uniqueness of the Standard Modelsp. 160
The Structure of ML[subscript 0]p. 173
The Topology of ML[subscript 0] and PL[subscript 0]p. 174
The Symplectic Structure of ML[subscript 0]p. 182
Topological Equivalencep. 188
Duality and Tangential Coordinatesp. 191
Epiloguep. 204
Addendum The Action of Mapping Classes on ML[subscript 0]p. 210
Bibliographyp. 214
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691025315
ISBN-10: 0691025312
Series: Annals of Mathematics Studies (Paperback)
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 232
Published: 23rd December 1991
Country of Publication: US
Dimensions (cm): 23.5 x 15.52  x 1.52
Weight (kg): 0.33