Walton, Jamie Paul; (1997) Algebraic properties of surface fibrations. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Algebraically, surface fibrations correspond to extensions of surface groups via their long homotopy exact sequences. First, it is proved that any group can be constructed by at most finitely many group extensions where the kernel and quotient correspond to finite free products of free groups and surface groups. This rigidity theorem has the important corollary that the group of all automorphisms of an extension of surface groups has finite index in the automorphism group of the fundamental group of a surface fibration. The Baer-Nielsen theorem for surfaces is extended to show that the natural homomorphism from the homotopy classes of diffeomorphisms of surface fibrations maps surjectively onto the outer automorphism group of their fundamental group. The virtual cohomological dimension of the outer automorphism groups of poly-surface and poly-free groups is calculated when the image of the operator homomorphism of the extension is finite. Using pure diffeomorphisms, this dimension is obtained when the image of the operator homomorphism is generated by Dehn twists about separating circles in a surface. A bound is also given on the virtual dimension of the automorphism group in all cases. Finally, it is shown the mapping class group of a Stallings fibration M is not rigid in the sense that the automorphism group of the long homotopy exact sequence of M does not have finite index in the automorphism group of the fundamental group of M. The virtual cohomological dimension of the mapping class group of the trivial Stallings fibration is calculated to be 6g-5 where g is the genus of the fibre, whereas Stallings fibrations constructed from pseudo-Anosov diffeomorphisms are shown to have finite mapping class groups.

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