On the ideal triangulation graph of a punctured surface

Mustafa Korkmaz[1]; Athanase Papadopoulos[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.

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 4, page 1367-1382
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Korkmaz, Mustafa, and Papadopoulos, Athanase. "On the ideal triangulation graph of a punctured surface." Annales de l’institut Fourier 62.4 (2012): 1367-1382. <http://eudml.org/doc/251130>.

@article{Korkmaz2012,
abstract = {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.},
affiliation = {Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey.; 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.},
author = {Korkmaz, Mustafa, Papadopoulos, Athanase},
journal = {Annales de l’institut Fourier},
keywords = {mapping class group; surface; arc complex; ideal triangulation; ideal triangulation graph; curve complex; Gromov hyperbolic},
language = {eng},
number = {4},
pages = {1367-1382},
publisher = {Association des Annales de l’institut Fourier},
title = {On the ideal triangulation graph of a punctured surface},
url = {http://eudml.org/doc/251130},
volume = {62},
year = {2012},
}

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
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 4
SP - 1367
EP - 1382
AB - 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.
LA - eng
KW - mapping class group; surface; arc complex; ideal triangulation; ideal triangulation graph; curve complex; Gromov hyperbolic
UR - http://eudml.org/doc/251130
ER -

References

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  7. N. V. Ivanov, J. D. McCarthy, On injective homomorphisms between Teichmüller modular groups, I. Invent. Math. 135 (1999), 425-486 Zbl0978.57014MR1666775
  8. M. Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology and its Applications 95 (1999), 85-111 Zbl0926.57012MR1696431
  9. M. Korkmaz, A. Papadopoulos, On the arc and curve complex of a surface Zbl1194.57026
  10. F. Luo, Automorphisms of the complex of curves, Topology 39 (2000), 283-298 Zbl0951.32012MR1722024
  11. D. Margalit, Automorphisms of the pants complex, Duke Math. J. 121 (2004), 457-479 Zbl1055.57024MR2040283
  12. R. C. Penner, The decorated Teichmüller space of punctured surfaces, Communications in Mathematical Physics 113 (1987), 299-339 Zbl0642.32012MR919235

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