Cofinitely Hopfian groups, open mappings and knot complements

TL;DR

This paper investigates cofinitely Hopfian groups, characterizes their properties in relation to knot groups and free-by-cyclic groups, and explores applications to open mappings between manifolds.

Contribution

It provides new characterizations of cofinitely Hopfian groups, especially in the context of knot groups and free-by-cyclic groups, and links these properties to geometric and topological applications.

Findings

Knot group is cofinitely Hopfian iff the knot is not a torus knot.

A free-by-cyclic group is cofinitely Hopfian iff it has trivial centre.

Applications to open mappings between manifolds are established.

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.