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, pp. 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 : 10.5802/aif.2725
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.
Korkmaz, Mustafa 1 ; Papadopoulos, Athanase 2

1 Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey.
2 Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany Institut de Recherche Mathématique Avancée, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France.
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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/articles/10.5802/aif.2725/

[1] Gromov, M. Hyperbolic groups, Essays in Group Theory, edited by S.M. Gersten (MSRI Publications 8), Springer-Verlag, 1987, pp. 75-263 | MR | Zbl

[2] Harer, J. L. Stability of the homology of the mapping class groups of orientable surfaces, Annals of Math., Volume 121 (1985), pp. 215-249 | DOI | MR | Zbl

[3] Hatcher, A. On triangulations of surfaces, Top. and its Appl., Volume 41 (1991), pp. 189-194 (A new version is available in the author’s webpage) | DOI | MR | Zbl

[4] Irmak, E.; Korkmaz, M. Automorphisms of the Hatcher-Thurston complex, Isr. J. Math., Volume 162 (2007), pp. 183-196 | DOI | MR | Zbl

[5] Irmak, E.; McCarthy, J. D. Injective simplicial maps of the arc complex, Turkish Journal of Mathematics, Volume 33 (2009), pp. 1-16 | MR | Zbl

[6] Ivanov, N. V. Automorphisms of Teichmüller modular groups (Lecture Notes in Math.), Springer-Verlag, Berlin and New York, 1988 no. 1346, pp. 199-270 | MR | Zbl

[7] Ivanov, N. V.; McCarthy, J. D. On injective homomorphisms between Teichmüller modular groups, I. Invent. Math., Volume 135 (1999) no. 2, pp. 425-486 | DOI | MR | Zbl

[8] Korkmaz, M. Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology and its Applications, Volume 95 (1999) no. 2, pp. 85-111 | DOI | MR | Zbl

[9] Korkmaz, M.; Papadopoulos, A. On the arc and curve complex of a surface (Math. Proc. Cambridge Philos. Soc., to appear.)

[10] Luo, F. Automorphisms of the complex of curves, Topology, Volume 39 (2000) no. 2, pp. 283-298 | DOI | MR | Zbl

[11] Margalit, D. Automorphisms of the pants complex, Duke Math. J., Volume 121 (2004) no. 3, pp. 457-479 | DOI | MR | Zbl

[12] Penner, R. C. The decorated Teichmüller space of punctured surfaces, Communications in Mathematical Physics, Volume 113 (1987), pp. 299-339 | DOI | MR | Zbl

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