Cofinitely Hopfian groups, open mappings and knot complements

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

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 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.