**Authors:** Andrei Jaikin-Zapirain (ICMAT-UAM)

**Source:** *Geometry & Topology*

**Date of publication:** 25 March, 2020

**Review:**

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 *S*¹. 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.