Measured geodesic laminations in Flatland
Thomas Morzadec1
1 Département de Mathématique UMR 8628 CNRS Université Paris-Sud Bât 430, Bureau 16 F-91405 Orsay Cedex (France)
Séminaire de théorie spectrale et géométrie, Volume 31 (2012-2014), pp. 117-136.

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 (that is a singular flat surface with holonomy {±Id}), 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.

DOI: 10.5802/tsg.297
Classification: 30F30, 53C12, 53C22
Keywords: Measured geodesic lamination, surface, half-translation structure, holomorphic quadratic differential, measured foliation, hyperbolic surface, dual tree
Thomas Morzadec 1

1 Département de Mathématique UMR 8628 CNRS Université Paris-Sud Bât 430, Bureau 16 F-91405 Orsay Cedex (France) 
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Thomas Morzadec. Measured geodesic laminations in Flatland. Séminaire de théorie spectrale et géométrie, Volume 31 (2012-2014), pp. 117-136. doi : 10.5802/tsg.297. https://tsg.centre-mersenne.org/articles/10.5802/tsg.297/

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