Train track

A simplicial graph $\tau$ imbedded in a differentiable surface $S$ (cf. also Differentiable manifold; Graph) with the property that the edges abutting on any given vertex have a common tangent there. In a neighbourhood of each vertex, the edges are divided into two non-empty classes, the vertices abutting "from one side" and the vertices abutting "from the other side". A vertex of a train track is also called a switch. The complementary regions of the train track in $S$ are surfaces which can have cusps on their boundary, and it is assumed that these complementary regions are not discs with zero or one cusp. This condition is important for the definition of measured foliations using train tracks (the construction is defined below). In practice, it suffices to deal with trivalent train tracks, that is, train tracks where the degree at each vertex is three (that is, there is one incoming edge from one side and two incoming edges from the other side).

A weighted train track $(\tau,\mu)$ is a train track $\tau$ together with a rule $\mu$ which assigns a non-negative real number to each edge of $\tau$ (this number being called the weight of that edge), with the property that at each switch, the sum of the weights of the edges abutting from one side is equal to the sum of the weights of the edges abutting from the other side. A weighted train track $(\tau,\mu)$ such that $\mu$ is not identically zero defines an equivalence class of measured foliations on $S$, by the following construction: Each edge of $\tau$ which has non-zero weight is replaced by a rectangle imbedded in $S$ as a regular neighbourhood of that edge and equipped with the standard foliation by leaves which are (nearly) parallel to the given edge, equipped with a transverse measure whose total mass is equal to the weight of the edge. These foliated rectangles are glued together by measure-preserving mappings, in a way which is naturally indicated by the way in which the edges of the train track $\tau$ fit together. Each complementary component of the foliated region is then collapsed to a spine, and the result is a measured foliation on $S$ which is well-defined up to isotopy and Whitehead moves. This measured foliation is said to be "carried by" the train track $\tau$. Note that this construction works also if the notion of "measured foliation" is replaced by that of "measured lamination", and the two points of view (laminations and foliations on surfaces) are equivalent (cf. also Lamination). Note also that the construction shows that the (weighted) train track is in a sense a quotient space of the (measured) foliation.

Train tracks have been introduced by W. Thurston in [a3], and they have been extensively used by Thurston and others in the study of the action of the mapping class group on a surface, and in the action of that group on the space of projective measured foliations. A comprehensive study of train tracks is done in [a1].

There exist higher-dimensional analogues of train tracks (called branched manifolds), which are useful in the study of surfaces and laminations in higher-dimensional manifolds (see [a2]). Note that a notion close to the notion of train track is already contained in [a4], although it is not used in the same manner.

References

[a1]	R.C. Penner, J.L. Harer, "Combinatorics of train tracks" , Ann. Math. Studies , 125 , Princeton Univ. Press (1992) MR1144770 Zbl 0765.57001
[a2]	W. Floyd, U. Oertel, "Incompressible surfaces via branched surfaces" Topology , 23 (1984) pp. 117–125 MR0721458 Zbl 0524.57008
[a3]	W.P. Thurston, "The Geometry and topology of three-manifolds" , Princeton Univ. Press (1978)
[a4]	R.F. Williams, "Expanding attractors" Inst. Hautes Etudes Sci. Publ. Math. , 43 (1974) pp. 169–203 MR0348794 Zbl 0338.58013 Zbl 0341.58011 Zbl 0279.58013 Zbl 0208.25801