Automorphisms of complexes of curves on punctured spheres and on punctured tori


TOPOLOGY AND ITS APPLICATIONS, vol.95, no.2, pp.85-111, 1999 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 95 Issue: 2
  • Publication Date: 1999
  • Doi Number: 10.1016/s0166-8641(97)00278-2
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.85-111
  • Keywords: mapping class group, complex of curves, surface, Teichmuller space
  • Middle East Technical University Affiliated: Yes


Let S be either a sphere with greater than or equal to 5 punctures or a torus with greater than or equal to 3 punctures. We prove that the automorphism group of the complex of curves of S is isomorphic to the extended mapping class group M-S*. As applications we prove that surfaces of genus less than or equal to 1 are determined by their complexes of curves, and any isomorphism between two subgroups of M-S(*) of finite index is the restriction of an inner automorphism of M-S(*) We conclude that the outer automorphism group of a finite index subgroup of M-S(*) is finite, extending the fact that the outer automorphism group of M-S(*) is finite. For surfaces of genus greater than or equal to 2, corresponding results were proved by Ivanov (MES/M/89/60, Preprint). (C) 1999 Elsevier Science B.V. All rights reserved.