Measured geodesic laminations in Flatland

TL;DR

This survey introduces flat laminations as a generalization of hyperbolic geodesic laminations on surfaces with half-translation structures, defining measures, topology, and classifying these laminations.

Contribution

It extends the concept of geodesic laminations to flat structures, defines transverse measures, and provides a classification theorem for measured flat laminations on surfaces.

Findings

The space of measured flat laminations is projectively compact.

A natural continuous projection from flat to hyperbolic measured laminations is constructed.

Most finite metric fat graphs support uncountably many measured flat laminations.

Abstract

Since their introduction by Thurston, measured geodesic laminations on hyperbolic surfaces occur in many contexts. In this survey, we give a generalization of geodesic laminations on surfaces endowed with a half-translation structure, called flat laminations, and we define transverse measures on flat laminations similar to transverse measures on hyperbolic laminations, taking into account that the images of the leaves of a flat lamination are in general not pairwise disjoint. One aim is to construct a tool that could allow a fine description of the space of degenerations of half-translation structures on a surface. We define a 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 any arbitrary half-translation structure and hyperbolic metric on a surface. We prove in particular that the space of measured flat laminations is projectively compact. The main result of this survey is a classification theorem of (measured) flat laminations on a compact surface endowed with a half-translation structure. We also give an exposition of that every finite metric fat graph, outside four homeomorphisms classes, is the support of uncountably many measured flat laminations with uncountably many leaves none of which is eventually periodic, and that the space of measured flat laminations is separable and projectively compact.