# ANNALES DE L'INSTITUT FOURIER

On the ideal triangulation graph of a punctured surface
Annales de l'Institut Fourier, Volume 62 (2012) no. 4, p. 1367-1382

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.

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.

Accepted : 2011-01-12
Classification:  32G15,  20F38,  30F10
Keywords: mapping class group ; surface ; arc complex ; ideal triangulation ; ideal triangulation graph ; curve complex ; Gromov hyperbolic.
@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},
zbl = {1256.32015},
mrnumber = {3025746},
language = {en},
url = {aif.centre-mersenne.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, Volume 62 (2012) no. 4, pp. 1367-1382. https://aif.centre-mersenne.org/item/AIF_2012__62_4_1367_0/

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