Nontorus knot groups are hyper-Hopfian

The group of any nontorus knot is hyperhopfian. A group G is called hopfian if every homomorphism from G onto itself is an automorphism. One can show using techniques of Nielsen that every finitely generated free group is hopfian. In 1930, H. Hopf [6] asked whether there exist finitely generated, and in particular, finitely presented groups that do not have this property. B. H. Neumann provided the first examples of finitely generated nonhopfian groups [9]. Finitely presented examples were described by G. Higman shortly afterwards [5]. Every finitely generated residually finite group is hopfian [8]. Since knot groups are residually finite [10], they are also hopfian. Recently, R. Daverman introduced the more general notion of hyperhopficity in work on approximate fibrations [1, 2]. A group G is hyperhopfian if every homomorphism y/:G-*G, with y/(G) < \ G and G/y/(G) cyclic, is an automorphism. Any hyperhopfian group is hopfian, but not conversely. Since the exterior of any torus knot is a nontrivial finite cyclic coverin