Automorphisms of surface mapping class groups. A recent theorem of N. Ivanov

Abstract

Ivanov of the Leningrad Branch of the Steklov Mathematical Institute has shown that the outer automorphism groups of surface mapping class groups are finite. In this report, we shall give explicit descriptions of the outer automorphism groups of both the mapping class groups and extended mapping class groups of closed, connected, orientable surfaces of negative Euler characteristic. (The extended mapping class groups are the extensions of the mapping class groups by the orientation reversing isotopy classes.) For surfaces of genus greater than two, these are, respectively, a cyclic group of order two and the trivial group. For a surface of genus two, these are both noncyclic groups of order four. (Once again, the hypergeometric involution in genus two plays a unique role.)