Groups, Geometry, and Dynamics

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Volume 4, Issue 4, 2010, pp. 693–707
DOI: 10.4171/GGD/101

Cofinitely Hopfian groups, open mappings and knot complements

Martin R. Bridson[1], Daniel Groves[2], Jonathan A. Hillman[3] and Gaven J. Martin[4]

(1) Mathematical Institute, University of Oxford, 24-29 St Giles', OX1 3LB, Oxford, UK
(2) Department of Mathematics, Statistics & Computer S, University of Illinois at Chicago, 851 S. Morgan St., IL 60607-7045, Chicago, USA
(3) School of Mathematics and Statistics F07, The University of Sydney, NSW 2006, Sydney, Australia
(4) Institute of Information and Mathematical Sciences, Massey University, Albany Campus, Private Bag 102-904, Auckland, New Zealand

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.

Keywords: Cofinitely Hopfian, open mappings, relatively hyperbolic, free-by-cyclic, knot groups

Bridson Martin, Groves Daniel, Hillman Jonathan, Martin Gaven: Cofinitely Hopfian groups, open mappings and knot complements. Groups Geom. Dyn. 4 (2010), 693-707. doi: 10.4171/GGD/101