Measured geodesic laminations in Flatland
Séminaire de théorie spectrale et géométrie, Tome 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
Mots clés : Measured geodesic lamination, surface, half-translation structure, holomorphic quadratic differential, measured foliation, hyperbolic surface, dual tree
Morzadec, Thomas 1

1 Département de Mathématique UMR 8628 CNRS Université Paris-Sud Bât 430, Bureau 16 F-91405 Orsay Cedex (France)
@article{TSG_2012-2014__31__117_0,
     author = {Morzadec, Thomas},
     title = {Measured geodesic laminations in {Flatland}},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {117--136},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {31},
     year = {2012-2014},
     doi = {10.5802/tsg.297},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/tsg.297/}
}
TY  - JOUR
AU  - Morzadec, Thomas
TI  - Measured geodesic laminations in Flatland
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2012-2014
SP  - 117
EP  - 136
VL  - 31
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/tsg.297/
DO  - 10.5802/tsg.297
LA  - en
ID  - TSG_2012-2014__31__117_0
ER  - 
%0 Journal Article
%A Morzadec, Thomas
%T Measured geodesic laminations in Flatland
%J Séminaire de théorie spectrale et géométrie
%D 2012-2014
%P 117-136
%V 31
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/tsg.297/
%R 10.5802/tsg.297
%G en
%F TSG_2012-2014__31__117_0
Morzadec, Thomas. Measured geodesic laminations in Flatland. Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 117-136. doi : 10.5802/tsg.297. http://archive.numdam.org/articles/10.5802/tsg.297/

[1] Bonahon, Francis The geometry of Teichmüller space via geodesic currents, Invent. Math., Volume 92 (1988) no. 1, pp. 139-162 | DOI | MR | Zbl

[2] Bonahon, Francis Geodesic laminations on surfaces, Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998) (Contemp. Math.), Volume 269, Amer. Math. Soc., Providence, RI, 2001, pp. 1-37 | DOI | MR | Zbl

[3] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319, Springer-Verlag, Berlin, 1999, pp. xxii+643 | DOI | MR | Zbl

[4] Duchin, Moon; Leininger, Christopher J.; Rafi, Kasra Length spectra and degeneration of flat metrics, Invent. Math., Volume 182 (2010) no. 2, pp. 231-277 | DOI | MR | Zbl

[5] Marden, Albert; Strebel, Kurt On the ends of trajectories, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 195-204 | MR | Zbl

[6] Morgan, John W.; Shalen, Peter B. Free actions of surface groups on R-trees, Topology, Volume 30 (1991) no. 2, pp. 143-154 | DOI | MR | Zbl

[7] Morgan, John W.; Shalen, Peter B. Free actions of surface groups on R-trees, Topology, Volume 30 (1991) no. 2, pp. 143-154 | DOI | MR | Zbl

[8] Morzadec, Thomas Laminations géodésiques plates (http://arxiv.org/abs/1311.7586)

[9] Morzadec, Thomas Measured flat geodesic laminations (http://arxiv.org/abs/1412.1994)

[10] Otal, Jean-Pierre Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math. (2), Volume 131 (1990) no. 1, pp. 151-162 | DOI | MR | Zbl

[11] Strebel, Kurt Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 5, Springer-Verlag, Berlin, 1984, pp. xii+184 | DOI | MR | Zbl

[12] Wolff, Maxime Connected components of the compactification of representation spaces of surface groups, Geom. Topol., Volume 15 (2011) no. 3, pp. 1225-1295 | DOI | MR | Zbl

Cité par Sources :