On the arc and curve complex of a surface


Creative Commons License

KORKMAZ M. , Papadopoulos A.

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, cilt.148, ss.473-483, 2010 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 148
  • Basım Tarihi: 2010
  • Doi Numarası: 10.1017/s0305004109990387
  • Dergi Adı: MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY
  • Sayfa Sayıları: ss.473-483

Özet

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.