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

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.

22 pages, 3 figures, This paper is based on a lecture at MSJ-KMS Joint Meeting 2023. To appear in Springer Proc. Math. Stat., "Proceedings of the MSJ-KMS Joint Meeting 2023''. This paper is a companion to the paper: "Dehn filling and the knot group I: Realization property'' [Int. Math. Res. Not. IMRN, 2025, 1-31]