Laminations géodésiques plates  [ Flat geodesic laminations ]
Annales de l'Institut Fourier, Volume 66 (2016) no. 1, p. 105-141

Since their introduction by Thurston, geodesic laminations on hyperbolic surfaces occur in many contexts. In this paper, we propose a generalization of geodesic laminations on locally CAT(0), complete, geodesic metric spaces, whose boundary at infinity of the universal cover is endowed with an invariant total cyclic order. Then we study these new objects on surfaces endowed with half-translation structures and on finite metric graphs. The main result of the paper is a theorem of classification of geodesic laminations on a compact surface endowed with a half-translation structure. We also show that every finite connected metric fat graph, without extremal point, outside four homeomorphism classes, is the support of a geodesic lamination with uncountably many leaves none of which is eventually periodic.

Depuis leur introduction par Thurston, les laminations géodésiques sur les surfaces hyperboliques interviennent dans de nombreux domaines. Dans cet article, on introduit une généralisation des laminations géodésiques sur les espaces métriques complets, géodésiques, localement CAT(0), tels que le bord à l’infini de leur revêtement universel est muni d’un ordre cyclique, invariant par l’action du groupe de revêtement. On étudie ces nouveaux objets sur les surfaces munies de structures de demi-translation et sur les graphes (métriques) finis. Le résultat principal de l’article est un théorème de classification des laminations géodésiques sur les surfaces compactes munies de structures de demi-translation. On démontre aussi que tous les graphes (métriques) finis, connexes, enrubannés, sans sommet terminal, hormis quatre (à homéomorphisme près), sont le support d’au moins une lamination géodésique plate avec une infinité indénombrable de feuilles, dont aucune n’est périodique à partir d’un certain temps.

Received : 2013-07-18
Revised : 2015-01-27
Accepted : 2015-02-24
Published online : 2016-02-17
DOI : https://doi.org/10.5802/aif.3007
Classification:  57M50,  51M05,  57R30,  30F60,  30F30,  30F35
Keywords: geodesic lamination, half-translation structure on surfaces, holomorphic quadratic differential, foliations with prong singularities, hyperbolic surface
@article{AIF_2016__66_1_105_0,
     author = {Morzadec, Thomas},
     title = {Laminations g\'eod\'esiques plates},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {1},
     year = {2016},
     pages = {105-141},
     doi = {10.5802/aif.3007},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2016__66_1_105_0}
}
Morzadec, Thomas. Laminations géodésiques plates. Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 105-141. doi : 10.5802/aif.3007. http://www.numdam.org/item/AIF_2016__66_1_105_0/

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