On the cyclic 3-manifold covers of the type surface x R

TL;DR

This paper proves that automorphism groups of certain cyclic 3-manifold covers act with compact quotients, extending fibered manifold theorems and establishing irreducibility of specific 3-manifold summands.

Contribution

It provides new proofs of automorphism group actions, fibered structures, and irreducibility results for specific classes of 3-manifolds, extending previous theorems and correcting earlier results.

Findings

Automorphism groups act with compact quotients under certain conditions.

Extension of fibered manifold theorems to new classes of 3-manifolds.

Proof of irreducibility of summands in torus sum decompositions.

Abstract

This article contains a proof of the fact that, under certain mild technical conditions, the action of the automorphism group of a cyclic 3-manifold cover of the type SxR, where S is a compact surface, yields a compact quotient. This result is then immediately applied to extend a theorem on the fiberings over the circle of certain compact 3-manifolds which are torus sums. As a corollary, I prove the validity of the conditional main theorem in my article titled "A fibering theorem for 3-manifolds", which appeared in the Journal of Groups, Complexity, Cryptology in 2021 and its subsequent erratum. This paper also furnishes a proof of the irreducibility of the summands of compact 3-manifolds which are torus sums and irreducible, and a proof of the interesting observation that a compact connected orientable 3-manifold whose fundamental group contains a subnormal subgroup of infinite index is the connected sum of an irreducible compact 3-manifold with (finitely many) 3-balls.