Combinatorics of Train Tracks

by Robert C. Penner, John L. Harer

Overview

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.

Product Details

ISBN-13:	9780691025315
Publisher:	Princeton University Press
Publication date:	12/23/1991
Series:	Annals of Mathematics Studies , #125
Pages:	232
Product dimensions:	6.00(w) x 9.25(h) x (d)

Table of Contents

174

	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]
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

Editorial Reviews

"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 fòr Mathematik

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 fòr Mathematik

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 fur Mathematik