[Flat geodesic laminations]
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 , 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 , 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.
Revised:
Accepted:
Published online:
Mot clés : lamination géodésique, surface munie d’une structure de demi-translation, différentielle quadratique holomorphe, feuilletage singulier, surface hyperbolique
Keywords: geodesic lamination, half-translation structure on surfaces, holomorphic quadratic differential, foliations with prong singularities, hyperbolic surface
Morzadec, Thomas 1
@article{AIF_2016__66_1_105_0, author = {Morzadec, Thomas}, title = {Laminations g\'eod\'esiques plates}, journal = {Annales de l'Institut Fourier}, pages = {105--141}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {66}, number = {1}, year = {2016}, doi = {10.5802/aif.3007}, language = {fr}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3007/} }
TY - JOUR AU - Morzadec, Thomas TI - Laminations géodésiques plates JO - Annales de l'Institut Fourier PY - 2016 SP - 105 EP - 141 VL - 66 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3007/ DO - 10.5802/aif.3007 LA - fr ID - AIF_2016__66_1_105_0 ER -
Morzadec, Thomas. Laminations géodésiques plates. Annales de l'Institut Fourier, Volume 66 (2016) no. 1, pp. 105-141. doi : 10.5802/aif.3007. https://aif.centre-mersenne.org/articles/10.5802/aif.3007/
[1] Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal., Volume 7 (1997) no. 2, pp. 215-244
[2] Geodesic currents on negatively curved groups, Arboreal group theory (Math. Sci. Res. Inst. Publ.), Volume 19, Springer, 1991, pp. 143-168 | DOI
[3] Geodesic laminations on surfaces, Laminations and foliations in dynamics, geometry and topology (Contemp. Math.), Volume 269, Amer. Math. Soc., Providence, RI, 2001, pp. 1-37 | DOI
[4] Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, 1999 | DOI
[5] Fundamentals of hyperbolic manifolds : Selected expositions (Canary, Richard; Epstein, David; Marden, Albert, eds.), Reprinted from the series London Mathematical Society Lecture Note Series 111(1986) and 112(1987), 2006
[6] Non-unique ergodicity, observers’ topology and the dual algebraic lamination for -trees, Illinois J. Math., Volume 51 (2007) no. 3, pp. 897-911 http://projecteuclid.org/euclid.ijm/1258131109
[7] On the large-scale geometry of flat surfaces, (Diss. 2010), 2011
[8] Length spectra and degeneration of flat metrics, Invent. Math., Volume 182 (2010) no. 2, pp. 231-277 | DOI
[9] Geometry of the mapping class groups. I. Boundary amenability, Invent. Math., Volume 175 (2009) no. 3, pp. 545-609 | DOI
[10] Foliations and laminations on hyperbolic surfaces, Topology, Volume 22 (1983) no. 2, pp. 119-135 | DOI
[11] On the ends of trajectories, Differential geometry and complex analysis, Springer, 1985, pp. 195-204
[12]
(Thèse de doctorat en préparation)[13] Combinatorics of train tracks, Annals of Mathematics Studies, 125, Princeton University Press, 1992
[14] Arbres, amalgames, ., 46, Astérisque, 1977
[15] Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 5, Springer-Verlag, 1984 | DOI
[16] Connected components of the compactification of representation spaces of surface groups, Geom. Topol., Volume 15 (2011) no. 3, pp. 1225-1295 | DOI
Cited by Sources: