Combinatorics Of Train Tracks. (am-125) (annals Of Mathematics Studies)
by R. C. Penner /
1992 / English / DjVu
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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.
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.
