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Combinatorics of train tracks. (English) Zbl 0765.57001
Annals of Mathematics Studies. 125. Princeton, NJ: Princeton University Press. xi, 216 p. (1992).
This book contains a thorough development of the theory of train tracks in 2-manifolds. Train tracks are objects invented by Thurston to study collections of disjoint closed curves and other important structures in 2-manifolds. A train track is topologically a 1-dimensional graph imbedded in the surface, such that the edges are smooth arcs and at each vertex all edges share a single tangent line; moreover at least one edge emanates from each of the two possible tangent directions at the vertex. An assignment of nonnegative integral weights to the edges at a vertex, so that the total weights in each of the two directions are equal (the “switch conditions”) determines a collection of disjoint simple closed curves where each edge of the train track is replaced by the number of arcs specified by the weight; thus a train track may parameterize many collections. An assignment of possibly nonintegral weights is also meaningful; it determines a set of disjoint geodesics called a measured lamination. The measure is on arcs transverse to the geodesics. The space of all measured laminations is a completion of the space of collections of closed curves rather analogous to the completion of the integral lattice points to a Euclidean space; in particular, if one projectivizes by identifying two (nonzero) measured laminations whose measures are scalar multiples of each other, then the measured laminations arising from collections of simple closed curves are dense. The collection of weights on a train track (satisfying suitable technical conditions) parameterizes a region of the space of measured laminations, providing manifold coordinate charts on the spaces of measured laminations and projective measured laminations.
There are three major chapters. The first contains the basic combinatorial theory of train tracks. The technical concepts of recurrence and transverse recurrence are developed, and the construction of a measured lamination from a weighted train track is detailed. This chapter closes with a discussion of a generalization for surfaces with boundary (“train tracks with stops”). The second chapter examines combinatorial manipulations of train tracks such as splitting and shifting. Standard models (with respect to a fixed pair-of-pants decomposition of the surface) of train tracks are constructed and studied. The third chapter uses the theory developed in the first two chapters to study the spaces of measured laminations and projective laminations. A description of how the coordinate charts for the standard models fit together shows that the manifold structure on the space of measured laminations (respectively, projective measured laminations) is piecewise integral linear (respectively, piecewise projectively integral linear). This also provides a natural symplectic pairing on the tangential structure of the space of measured laminations which is described quite explicitly in terms of the chart coordinates. Additional addenda sketch the relation of measured laminations to measured foliations and describe the action of the surface mapping class group on the space of measured laminations.
The book is beautifully written, with a clear path of theoretical development amid a wealth of detail for the technician. Although most of the major results have appeared in published work of the authors and others, this text provides a valuable reference work as well as a readable introduction for the student or newcomer to the area.

MSC:
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57M50 General geometric structures on low-dimensional manifolds
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F60 Teichm├╝ller theory for Riemann surfaces
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