Recognition of being fibered for compact 3–manifolds

Authors: Andrei Jaikin-Zapirain (ICMAT-UAM)

Source: Geometry & Topology

Date of publication: 25 March, 2020



Much attention has been paid recently to those properties of 3-manifolds which can be deduced from the finite quotients of their fundamental groups; or, from another viewpoint, from the structure of their lattice of finite-sheeted coverings. Having assembled these finite quotients into the profinite completion of the fundamental group, this amounts to the study of “profinite invariants” of the 3-manifold. A profinite invariant may be defined as some property P of a group G such that, whenever H is a group with property P and  , then G also has property P. One may restrict attention to a particular class of groups (for example fundamental groups of compact orientable 3-manifolds) and require both G and H to be from that class.

A compact orientable 3-manifold M is fibered if M admits the structure of a surface bundle over S¹. Work of Stallings, together with the resolution of the Poincaré Conjecture, shows that M is fibered if the fundamental group π₁(M) of M is isomorphic to a semidirect product  N with N finitely generated. Thus, the fundamental group of a compact orientable 3-manifold detects whether the 3-manifold is fibered. We may ask whether to be fibered is also a profinite property. It is confirmed in this paper.

Theorem: Let M and N be two compact orientable 3-manifolds such that . Then M fibers over the circle if and only if N does.

The proof uses in an essential way results of I. Agol, P. Przytycki and D. Wise on separability of 3-manifold groups which has had significant implications on our understanding of the profinite completion of the fundamental groups of compact 3-manifolds.