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 ), 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.
Keywords: Measured geodesic lamination, surface, half-translation structure, holomorphic quadratic differential, measured foliation, hyperbolic surface, dual tree
@article{TSG_2012-2014__31__117_0,
author = {Thomas Morzadec},
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 = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.297/}
}
TY - JOUR AU - Thomas Morzadec 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 - https://proceedings.centre-mersenne.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 Thomas Morzadec %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 https://proceedings.centre-mersenne.org/articles/10.5802/tsg.297/ %R 10.5802/tsg.297 %G en %F TSG_2012-2014__31__117_0
Thomas Morzadec. Measured geodesic laminations in Flatland. Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 117-136. doi : 10.5802/tsg.297. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.297/
[1] The geometry of Teichmüller space via geodesic currents, Invent. Math., Volume 92 (1988) no. 1, pp. 139-162 | DOI | MR | Zbl
[2] 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] 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] Length spectra and degeneration of flat metrics, Invent. Math., Volume 182 (2010) no. 2, pp. 231-277 | DOI | MR | Zbl
[5] On the ends of trajectories, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 195-204 | MR | Zbl
[6] Free actions of surface groups on -trees, Topology, Volume 30 (1991) no. 2, pp. 143-154 | DOI | MR | Zbl
[7] Free actions of surface groups on -trees, Topology, Volume 30 (1991) no. 2, pp. 143-154 | DOI | MR | Zbl
[8] Laminations géodésiques plates (http://arxiv.org/abs/1311.7586)
[9] Measured flat geodesic laminations (http://arxiv.org/abs/1412.1994)
[10] 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] 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] 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 :