On the arc and curve complex of a surface

Korkmaz, Mustafa
Papadopoulos, Athanase
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.

Citation Formats
M. Korkmaz and A. Papadopoulos, “On the arc and curve complex of a surface,” MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, vol. 148, pp. 473–483, 2010, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/34500.