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Nontorus knot groups are hyperhopfian. (English) Zbl 0836.20049
A group $$G$$ is hopfian if every homomorphism from $$G$$ onto itself is an automorphism. Knot groups are well-known examples. Recently R. Daverman introduced the more general notion of hyperhopficity in work on approximate fibrations. A group $$G$$ is hyperhopfian if every homomorphism $$\psi:G\to G$$, with $$\psi(G)\triangleleft G$$ and $$G/\psi(G)$$ cyclic, is an automorphism. Any hyperhopfian group is hopfian, but not conversely. Since the exterior of any torus knot is a nontrivial finite cyclic covering space of itself, torus knot groups are nonhyperhopfian. In view of this, R. J. Daverman asked [in Indiana Univ. Math. J. 40, 1451- 1470 (1991; Zbl 0739.57007)] whether any other knot groups fail to be hyperhopfian. F. González-Acuña and W. Whitten went far towards providing a negative answer using topological methods. In this paper we use purely group theoretical arguments in order to establish the following Theorem: The group of every nontorus knot is hyperhopfian.

MSC:
 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 20E36 Automorphisms of infinite groups 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010) 57M10 Covering spaces and low-dimensional topology
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