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.
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.
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
@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}, pages = {1367--1382}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {62}, number = {4}, year = {2012}, doi = {10.5802/aif.2725}, mrnumber = {3025746}, zbl = {1256.32015}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2725/} }
TY - JOUR AU - Korkmaz, Mustafa AU - Papadopoulos, Athanase TI - On the ideal triangulation graph of a punctured surface JO - Annales de l'Institut Fourier PY - 2012 SP - 1367 EP - 1382 VL - 62 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2725/ DO - 10.5802/aif.2725 LA - en ID - AIF_2012__62_4_1367_0 ER -
%0 Journal Article %A Korkmaz, Mustafa %A Papadopoulos, Athanase %T On the ideal triangulation graph of a punctured surface %J Annales de l'Institut Fourier %D 2012 %P 1367-1382 %V 62 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2725/ %R 10.5802/aif.2725 %G en %F 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. doi : 10.5802/aif.2725. https://aif.centre-mersenne.org/articles/10.5802/aif.2725/
[1] Hyperbolic groups, Essays in Group Theory, edited by S.M. Gersten (MSRI Publications 8), Springer-Verlag, 1987, pp. 75-263 | MR | Zbl
[2] 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] 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] Automorphisms of the Hatcher-Thurston complex, Isr. J. Math., Volume 162 (2007), pp. 183-196 | DOI | MR | Zbl
[5] Injective simplicial maps of the arc complex, Turkish Journal of Mathematics, Volume 33 (2009), pp. 1-16 | MR | Zbl
[6] Automorphisms of Teichmüller modular groups (Lecture Notes in Math.), Springer-Verlag, Berlin and New York, 1988 no. 1346, pp. 199-270 | MR | Zbl
[7] On injective homomorphisms between Teichmüller modular groups, I. Invent. Math., Volume 135 (1999) no. 2, pp. 425-486 | DOI | MR | Zbl
[8] 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] On the arc and curve complex of a surface (Math. Proc. Cambridge Philos. Soc., to appear.)
[10] Automorphisms of the complex of curves, Topology, Volume 39 (2000) no. 2, pp. 283-298 | DOI | MR | Zbl
[11] Automorphisms of the pants complex, Duke Math. J., Volume 121 (2004) no. 3, pp. 457-479 | DOI | MR | Zbl
[12] The decorated Teichmüller space of punctured surfaces, Communications in Mathematical Physics, Volume 113 (1987), pp. 299-339 | DOI | MR | Zbl
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