Vanishing nontrivial elements in a knot group by Dehn fillings

Abstract

Let K be a nontrivial knot in S3 with the exterior E(K), and denote π1(E(K)) by G(K). We prove that for any hyperbolic knot K and any nontrivial element g∈G(K), there are only finitely many Dehn fillings of E(K) which trivialize g. We also demonstrate that there are infinitely many nontrivial elements in G(K) which cannot be trivialized by nontrivial Dehn fillings.