On the ideal triangulation graph of a punctured surface  [ Sur le graphe des triangulations idéales d’une surface épointée ]
Annales de l'Institut Fourier, Tome 62 (2012) no. 4, p. 1367-1382
On étudie le graphe T(S) des triangulations idéales d’une surface S orientée de type fini. On montre que si S n’est pas une sphère ayant au plus quatre perforations ou un tore ayant une seule perforation, l’application naturelle du groupe modulaire étendu de S dans le groupe d’automorphismes de T(S) est un isomorphisme. On montre aussi que le graphe T(S) d’une telle surface n’est pas hyperbolique au sens de Gromov. On montre enfin que si les graphe des triangulations idéales de deux surfaces orientées de type fini sont homéomorphes, alors les surfaces sont elles-mêmes homéomorphes.
We study the ideal triangulation graph T(S) of an oriented punctured surface S of finite type. We show that if S is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of S into the simplicial automorphism group of T(S) is an isomorphism. We also show that the graph T(S) of such a surface S, equipped with its natural simplicial metric is not Gromov hyperbolic. We also show that if the triangulation graph of two oriented punctured surfaces of finite type are homeomorphic, then the surfaces themselves are homeomorphic.
DOI : https://doi.org/10.5802/aif.2725
Classification:  32G15,  20F38,  30F10
Mots clés: groupe modulaire ; surface ; complexe des arcs ; triangulation idéale ; graphe des triangulations idéales ; complexe des courbes ; hyperbolicité au sens de Gromov.
@article{AIF_2012__62_4_1367_0,
     author = {Korkmaz, Mustafa and Papadopoulos, Athanase},
     title = {On the ideal triangulation graph of a punctured surface},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {4},
     year = {2012},
     pages = {1367-1382},
     doi = {10.5802/aif.2725},
     mrnumber = {3025746},
     zbl = {1256.32015},
     language = {en},
     url = {http://http://www.numdam.org/item/AIF_2012__62_4_1367_0}
}
Korkmaz, Mustafa; Papadopoulos, Athanase. On the ideal triangulation graph of a punctured surface. Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1367-1382. doi : 10.5802/aif.2725. http://www.numdam.org/item/AIF_2012__62_4_1367_0/

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