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

✍ Scribed by Mustafa Korkmaz

• Book ID: 104295390
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 305 KB
• Volume: 95
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(97)00278-2

✦ Synopsis

Let S be either a sphere with 5 punctures or a torus with 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 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 2, corresponding results were proved by Ivanov (IHES/M/89/60, Preprint).