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 treatment of the combinatorial structure of the space of measured geodesic laminations in a fixed surface.
Über den Autor
R. C. Penner & John L. Harer
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[subillegalscript 0]< | 173 |
| 3.1 | The Topology of ML[subillegalscript 0] and PL[subillegalscript 0]< | 174 |
| 3.2 | The Symplectic Structure of ML[subillegalscript 0]< | 182 |
| 3.3 | Topological Equivalence | 188 |
| 3.4 | Duality and Tangential Coordinates | 191 |
| Epilogue | 204 |
| Addendum The Action of Mapping Classes on ML[subillegalscript 0]< | 210 |
| Bibliography | 214 |
Klappentext
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.
