Dehn filling and the knot group III: cyclic persistent subgroups

Property P is equivalent to saying that for every non-trivial knot K , its meridian remains non-trivial for all non-trivial Dehn fillings. We call a non-trivial element g in the knot group G ( K ) persistent if it remains non-trivial for all non-trivial Dehn fillings. The purpose of this note is to show that K has no finite surgery if and only if G ( K ) contains infinitely many, mutually non-conjugate cyclic subgroups whose non-trivial elements are persistent.