Automorphism Subgroups of Finite Index in Algebraic Mapping Class Groups

We give an algebraic proof of the Birman-Bers theorem - an algebraic result whose previous proofs used topology or analysis, and which says that a certain subgroup of finite index in the (algebraic) mapping class group of an oriented punctured surface is isomorphic to a certain group of automorphisms. The index 2 case gives rise to an automorphism of the group consisting of those automorphisms of a free group that stabilize the normal subgroup generated by an oriented-surface relator, and we analyze this curious automorphism. © 1997 Academic Press.