Cofinitely Hopfian groups, open mappings and knot complements

Details Cofinitely Hopfian groups, open mappings and knot complements Cofinitely Hopfian groups, open mappings and knot complements

2010-12-08

arxiv.org/abs/1012.1785v1

Groups, Geometry and Dynamics, 4, no.4, 2010, 693-707

Mathematics

Group Theory

14 pages

Scientific paper

10.4171/GGD/101

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.

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