Cofinitely Hopfian groups, open mappings and knot complements

M.Bridson, D.Groves, J.A.Hillman, G.J.Martin

Abstract

A group $\mathrm{\Gamma }$ is defined to be cofinitely Hopfian if every homomorphism $\mathrm{\Gamma }\to \mathrm{\Gamma }$ 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 cofinitely Hopfian if and only if it has trivial centre. Applications to the theory of open mappings between manifolds are presented.

Keywords: Cofinitely Hopfian, open mappings, relatively hyperbolic, free-by-cyclic, knot groups.

AMS Subject Classification: Primary 20F65; secondary 57M25.