A group Γ is defined to be cofinitely Hopfian if every homomorphism Γ → Γ 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 cofinitely Hopfian if and only if it has trivial centre. Applications to the theory of open mappings between manifolds are presented.
Cite this article
Martin R. Bridson, Daniel Groves, Jonathan A. Hillman, Gaven J. Martin, Cofinitely Hopfian groups, open mappings and knot complements. Groups Geom. Dyn. 4 (2010), no. 4, pp. 693–707DOI 10.4171/GGD/101