Automorphism subgroups of finite index in algebraic mapping class groups
The mapping class group of an orientable surface acts on the Teichmüller space giving rise to the corresponding moduli space. Many investigations into mapping class groups have thus utilised topological and analytic information in this analysis. For punctured surfaces, the fundamental groups are free and these mapping class groups may be described as outer automorphism groups of purely algebraically defined subgroups of the automorphism group of a free group. In this paper, the authors give purely algebraic analyses and proofs of results which relate the mapping class group of an \(n+1\)-punctured surface to that of the corresponding \(n\)-punctured surface. In particular, they analyse the nature of the automorphism (classes) in the kernel of the surjective map from the pure mapping class group \(PM_{g,n+1}\to PM_{g,n}\), first described using Dehn twists by \textit{J. S. Birman} [Commun. Pure Appl. Math. 22, 213-238 (1969; Zbl 0167.21503)]. Their analysis is linked to the work of \textit{J. McCool} [Publ. Mat., Barc. 40, No. 2, 457-468 (1996)].
scientific article; zbMATH DE number 1013978
scientific article