Cofinitely Hopfian groups, open mappings and knot complements (with M. Bridson, J. Hillman and G. Martin). .pdf (172k)

Abstract 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.