Cofinitely Hopfian groups, open mappings and knot complements (with
J. Hillman and
A group $\Gamma$ is defined to be cofinitely Hopfian if every homomorphism
$\Gamma \to \Gamma$ whose image is of finite index is an automorphism. Geometrically
significant groups enjoying this property include certain relatively hyperbolic groups
and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus
knot. A free-by-cyclic group is confinitely Hopfian if and only if it has a trivial center.
Applications to the theory of open mappings between manifolds are presented.