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On the arc and curve complex of a surface

Published online by Cambridge University Press:  02 December 2009

MUSTAFA KORKMAZ  and
ATHANASE PAPADOPOULOS
MUSTAFA KORKMAZ
Affiliation:
Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey. e-mail: [email protected]
ATHANASE PAPADOPOULOS
Affiliation:
Institut de Recherche Mathématique Avancée, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France, and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. e-mail: [email protected]
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Abstract

We study the arc and curve complex AC(S) of an oriented connected surface S of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of AC(S) coincides with the natural image of the extended mapping class group of S in that group. We also show that for any vertex of AC(S), the combinatorial structure of the link of that vertex characterizes the type of a curve or of an arc in S that represents that vertex. We also give a proof of the fact if S is not a sphere with at most three punctures, then the natural embedding of the curve complex of S in AC(S) is a quasi-isometry. The last result, at least under some slightly more restrictive conditions on S, was already known. As a corollary, AC(S) is Gromov-hyperbolic.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 3 , May 2010 , pp. 473 - 483
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Hatcher, A.On triangulations of surfaces. Topology Appl. 40 (1991), 189–194. A new version is available on the author's webpage.CrossRefGoogle Scholar
[2]Harvey, W. J. Boundary structure of the modular group. Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference, pp. 245–251, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, NJ, 1981.Google Scholar
[3]Irmak, E. and McCarthy, J. D.Injective simplicial maps of the arc complex, Turkish J. Math. 33 (2009), 1–16.Google Scholar
[4]Ivanov, N. V. Automorphisms of complexes of curves and of Teichmüller spaces. Internat. Math. Res. Notices 1997, no. 14, 651–666.Google Scholar
[5]Korkmaz, M.Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topology Appl. 95, no. 2 (1999), 85–111.Google Scholar
[6]Luo, F.Automorphisms of the complex of curves. Topology 39, no. 2 (2000), 283–298.Google Scholar
[7]Masur, H. A. and Minsky, Y. N.Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138 (1) (1999), 103–149.CrossRefGoogle Scholar
[8]Schleimer, S. Notes on the complex of curves. Notes from a mini-course given at Caltech in January 2005, available on the web.Google Scholar