On étudie le graphe des triangulations idéales d’une surface orientée de type fini. On montre que si 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 dans le groupe d’automorphismes de est un isomorphisme. On montre aussi que le graphe 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 of an oriented punctured surface of finite type. We show that if 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 into the simplicial automorphism group of is an isomorphism. We also show that the graph of such a surface , 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.
@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},
volume = {62},
year = {2012},
pages = {1367-1382},
doi = {10.5802/aif.2725},
zbl = {1256.32015},
mrnumber = {3025746},
language = {en},
url = {http://dml.mathdoc.fr/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) pp. 1367-1382. doi : 10.5802/aif.2725. http://gdmltest.u-ga.fr/item/AIF_2012__62_4_1367_0/
[1] Hyperbolic groups, Essays in Group Theory, edited by S.M. Gersten, Springer-Verlag (MSRI Publications 8) (1987), pp. 75-263 | MR 919829 | Zbl 0634.20015
[2] Stability of the homology of the mapping class groups of orientable surfaces, Annals of Math., Tome 121 (1985), pp. 215-249 | Article | MR 786348 | Zbl 0579.57005
[3] On triangulations of surfaces, Top. and its Appl., Tome 41 (1991), pp. 189-194 (A new version is available in the author’s webpage) | Article | MR 1123262 | Zbl 0727.57012
[4] Automorphisms of the Hatcher-Thurston complex, Isr. J. Math., Tome 162 (2007), pp. 183-196 | Article | MR 2365859 | Zbl 1149.57032
[5] Injective simplicial maps of the arc complex, Turkish Journal of Mathematics, Tome 33 (2009), pp. 1-16 | MR 2681579 | Zbl 1206.57018
[6] Automorphisms of Teichmüller modular groups, Springer-Verlag, Berlin and New York (Lecture Notes in Math.) (1988) no. 1346, pp. 199-270 | MR 970079 | Zbl 0657.57004
[7] On injective homomorphisms between Teichmüller modular groups, I. Invent. Math., Tome 135 (1999) no. 2, pp. 425-486 | Article | MR 1666775 | Zbl 0978.57014
[8] Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology and its Applications, Tome 95 (1999) no. 2, pp. 85-111 | Article | MR 1696431 | Zbl 0926.57012
[9] On the arc and curve complex of a surface (Math. Proc. Cambridge Philos. Soc., to appear.)
[10] Automorphisms of the complex of curves, Topology, Tome 39 (2000) no. 2, pp. 283-298 | Article | MR 1722024 | Zbl 0951.32012
[11] Automorphisms of the pants complex, Duke Math. J., Tome 121 (2004) no. 3, pp. 457-479 | Article | MR 2040283 | Zbl 1055.57024
[12] The decorated Teichmüller space of punctured surfaces, Communications in Mathematical Physics, Tome 113 (1987), pp. 299-339 | Article | MR 919235 | Zbl 0642.32012