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

TL;DR

This paper demonstrates that for all nontrivial knots, there are infinitely many persistent elements in their groups that remain nontrivial under all nontrivial Dehn fillings, showing their structural pervasiveness.

Contribution

It proves the existence of infinitely many persistent elements with disjoint automorphic orbits in every nontrivial knot group, extending the understanding of their structural ubiquity.

Findings

Every nontrivial knot group has infinitely many persistent elements.

Persistent elements exist outside every proper subgroup of the knot group.

For certain hyperbolic knots, persistent elements are structurally pervasive.

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 $G(K)$.