Automorphism Subgroups of Finite Index in Algebraic Mapping Class Groups Original Research Article

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.