Dehn filling and the knot group II: Ubiquity of persistent elements

Abstract

Let $K$ be a nontrivial knot in $S^3$. We say that an element of the knot group $G(K)$ is \textit{persistent} if it remains nontrivial under all nontrivial Dehn fillings. Such elements exist for every nontrivial knot. Indeed, Property P is equivalent to the statement that the meridian of $K$ is a persistent element, and this represents the first instance of such elements. Building on the solution to the Property P conjecture due to Kronheimer and Mrowka, we show that every nontrivial knot group admits infinitely many persistent elements with pairwise disjoint automorphic orbits, none of which contains a power of the meridian. We then develop this further to show that for a broad class of hyperbolic knots - namely those admitting no surgery whose resulting manifold has torsion in its fundamental group - persistent elements are not rare curiosities, but rather structurally pervasive in $G(K)$. This is reflected in the following two properties: (i) Every subgroup of $G(K)$ that is not contained in the normal closure of a peripheral element contains persistent elements. (ii) Persistent elements exist outside every proper subgroup of $G(K)$.