Recognition of being fibered for compact $3$–manifolds

Abstract

Let $M$ be a compact orientable aspherical $3$–manifold. We show that if the profinite completion of ${\pi }_1\left(M\right)$ is isomorphic to the profinite completion of a free-by-cyclic group or to the profinite completion of a surface-by-cyclic group, then $M$ fibers over the circle with compact fiber.