Dehn filling and the knot group III: cyclic persistent subgroups
摘要
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.