JournalsggdVol. 4, No. 4pp. 693–707

Cofinitely Hopfian groups, open mappings and knot complements

  • Martin R. Bridson

    University of Oxford, UK
  • Daniel Groves

    University of Illinois at Chicago, United States
  • Jonathan A. Hillman

    The University of Sydney, Australia
  • Gaven J. Martin

    Massey University, Auckland, New Zealand
Cofinitely Hopfian groups, open mappings and knot complements cover
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Abstract

A group Γ is defined to be cofinitely Hopfian if every homomorphism Γ → Γ 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.

Cite this article

Martin R. Bridson, Daniel Groves, Jonathan A. Hillman, Gaven J. Martin, Cofinitely Hopfian groups, open mappings and knot complements. Groups Geom. Dyn. 4 (2010), no. 4, pp. 693–707

DOI 10.4171/GGD/101