Laminations, Foliations, and Trees

Abstract

AbstractIn this chapter, we discuss several essentially mathematical concepts: Geodesic currents with zero self-intersection. Measured foliations. Measured geodesic laminations. Trajectories of holomorphic quadratic differentials. Measured train tracks. Generalized interval exchange transformations. Small actions of surface groups on metric trees. There is yet another point of view on measured geodesic laminations [Hat88], which will not be discussed here. In this section, we describe spaces of measured foliations and geodesic laminations on the 2-torus S, how they compactify the Teichmüller space of S, and how elements of the mapping class group of S act on this compactification. It is helpful to keep in mind this description since it is the simplest version of what we will do in the case of hyperbolic surfaces.KeywordsQuadratic DifferentialMapping Class GroupHyperbolic SurfaceDehn TwistMeasured FoliationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.