Topology of 3-manifolds and a class of groups

TL;DR

This paper introduces a new class of groups inspired by 3-manifold topology, explores their properties, provides examples, and discusses implications for the virtual Betti number conjecture.

Contribution

It defines a novel class of groups related to 3-manifold coverings and proves foundational results and conjectures about these groups.

Findings

Identified classes of groups that belong or do not belong to this new class.

Proved elementary properties of these groups.

Connected the conjectures to Thurston's virtual Betti number conjecture.

Abstract

This paper grew out of an attempt to find a suitable finite sheeted covering of an aspherical 3-manifold so that the cover either has infinite or trivial first homology group. With this motivation we define a new class of groups. These groups are in some sense eventually perfect. We prove results giving several classes of examples of groups which do (not) belong to this class. Also we prove some elementary results on these groups and state two conjectures. A direct application of one of the conjectures to the virtual Betti number conjecture of Thurston is mentioned.