## 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.

: Primary 20F65; secondary 57M25.

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 Monday, August 16, 2010