Sufficient Conditions for Recognizing a 3-manifold Group

TL;DR

This paper investigates conditions under which a group can be recognized as a 3-manifold group, providing partial criteria and an invariant to measure how close a presentation is to originating from a 3-manifold.

Contribution

It introduces conditions and an invariant to identify when a group presentation arises from a 3-manifold, advancing understanding of the Isomorphism Problem.

Findings

Established partial conditions for recognizing 3-manifold groups

Defined an invariant measuring the proximity of a presentation to a 3-manifold group

Connected group presentations to Heegaard diagrams and 3-manifold fundamental groups

Abstract

In this work we ask when a group is a 3-manifold group, or more specifically, when does a group presentation come naturally from a Heegaard diagram for a 3-manifold? We will give some conditions for partial answers to this form of the Isomorphism Problem by addressing how the presentation associated to a diagram for a splitting is related to the fundamental group of a 3-manifold. In the process, we determine an invariant of groups (by way of group presentations) for how far such presentations are from 3-manifolds.