Rigidity of Fibering

Abstract

Given a manifold M, it is natural to ask in how many ways it fibers (we mean fibering in a general way, where the base might be an orbifold -this could be described as Seifert fibering)There are group-theoretic obstructions to the existence of even one fibering, and in some cases (such as Kähler manifolds or three-dimensional manifolds) the question reduces to a group-theoretic question. In this note we summarize the author's state of knowledge of the subject.I would like to thank Ilya Kapovich for much enlightenment on geometric group theory, D. Arapura for supplying the proof of Theorem 1.2, Igor Belegradek and Alain Valette on putting me abreast of the recent developments on groups with finite outer automorphism group, Fred Cohen for interesting discussions on different sorts of monodromy, N. Katz and P. Sarnak, who piqued the author's interest in monodromy, and the Institute for Advanced Study for its hospitality, which made this work possible. I would like to thank Igor Belegradek, Tom Church, and David Futer on comments on a previous version of this preprint -in particular, Belegradek had pointed out the very interesting preprint [22]. I would also like to thank