On the arc and curve complex of a surface

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2010-05-01

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Korkmaz, Mustafa

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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.

Subject Keywords

General Mathematics

URI

https://hdl.handle.net/11511/34500

Journal

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY

DOI

https://doi.org/10.1017/s0305004109990387

Collections

Department of Mathematics, Article

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Citation Formats

M. Korkmaz, “On the arc and curve complex of a surface,” MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, pp. 473–483, 2010, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/34500.