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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: korkmaz@metu.edu.tr
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: papadopoulos@math.u-strasbg.fr
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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

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