Measured flat geodesic laminations

Abstract

Since their introduction by Thurston, measured geodesic laminations on hyperbolic surfaces occur in many contexts. In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a half-translation structure (that is a singular flat surface with holonomy {+/-Id}, similar to geodesic laminations on hyperbolic surfaces. Here is a sequel to this article that aims at defining transverse measures on flat laminations similar to transverse measures on hyperbolic laminations, taking into account that two different leaves of a flat lamination may no longer be disjoint. One aim of this paper is to construct a tool that could allow a fine description of the space of degenerations of half-translation structures on a surface. In this paper, we define a nicer topology than the Hausdorff topology on the set of measured flat laminations and a natural continuous projection of the space of measured flat laminations onto the space of measured hyperbolic laminations, for some arbitrary half-translation structure and hyperbolic metric on a surface. We prove in particular that the space of measured flat laminations is projectively compact.