Measured flat geodesic laminations

TL;DR

This paper extends the concept of measured geodesic laminations to flat laminations on surfaces with half-translation structures, introducing a new topology and a projection to hyperbolic laminations, aiding the study of degenerations.

Contribution

It defines transverse measures on flat laminations, constructs a topology for measured flat laminations, and establishes a continuous projection to hyperbolic laminations.

Findings

Introduces a topology on measured flat laminations.

Defines a continuous projection onto hyperbolic laminations.

Proves the space of measured flat laminations is projectively compact.

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.