Group theoretic perspective on Dehn fillings: Property P conjecture and beyond

TL;DR

This paper explores Dehn fillings of 3-manifolds using group theory, building on the Property P conjecture, and offers new insights into the topological implications of these fillings.

Contribution

It introduces a group theoretic framework for studying Dehn fillings and extends the Property P conjecture to new variations, providing fresh perspectives.

Findings

New group theoretic approaches to Dehn fillings

Extensions of the Property P conjecture

Potential implications for 3-manifold topology

Abstract

The Property P Conjecture, which was settled by Kronheimer and Mrowka, asserts that every $3$--manifold obtained by non-trivial Dehn surgery on a non-trivial knot is never simply connected. We propose new perspectives in studying Dehn filling from group theoretic point of view, which stem from several variation of the Property P conjecture.