On the ideal triangulation graph of a punctured surface

Korkmaz, Mustafa; Papadopoulos, Athanase

Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1367-1382.

On étudie le graphe $T\left(S\right)$ 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 $T\left(S\right)$ est un isomorphisme. On montre aussi que le graphe $T\left(S\right)$ 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 $T\left(S\right)$ 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 $T\left(S\right)$ is an isomorphism. We also show that the graph $T\left(S\right)$ 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.

Classification : 32G15, 20F38, 30F10
Keywords: mapping class group, surface, arc complex, ideal triangulation, ideal triangulation graph, curve complex, Gromov hyperbolic.
Mot clés : groupe modulaire, surface, complexe des arcs, triangulation idéale, graphe des triangulations idéales, complexe des courbes, hyperbolicité au sens de Gromov.

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