Topology of 3-manifolds and a class of groups II

TL;DR

This paper investigates a new class of groups related to 3-manifold topology, providing examples, basic properties, and conjectures, with implications for the virtual Betti number conjecture.

Contribution

It introduces a new class of groups, explores their properties, and connects these findings to the virtual Betti number conjecture in 3-manifold topology.

Findings

Identified classes of groups belonging or not belonging to the new class

Proved basic properties of these groups

Formulated two conjectures related to the class

Abstract

This is a continuation of an earlier preprint (math.GT/0209121) under the same title. These papers 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 defined a new class of groups. These groups are in some sense eventually perfect. Here we prove results giving several classes of examples of groups which do (not) belong to this class. Also we prove some basic results on these groups and state two conjectures. A direct application of one of the conjectures to the virtual Betti number conjecture is mentioned. For completeness, here we reproduce parts of math.GT/0209121.