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Article Dans Une Revue Mathematical Proceedings of the Cambridge Philosophical Society Année : 2010

On the arc and curve complex of a surface

Résumé

We study the {\it 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.
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Dates et versions

hal-00405188 , version 1 (19-07-2009)
hal-00405188 , version 2 (21-07-2009)
hal-00405188 , version 3 (23-07-2009)
hal-00405188 , version 4 (11-08-2009)

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Mustafa Korkmaz, Athanase Papadopoulos. On the arc and curve complex of a surface. Mathematical Proceedings of the Cambridge Philosophical Society, 2010, 148 (3), ⟨10.1017/S0305004109990387⟩. ⟨hal-00405188v4⟩
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