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, UNITED KINGDOM
(2) Department of Mathematics, Statistics & Computer S, University of Illinois at Chicago, 851 S. Morgan St., IL 60607-7045, CHICAGO, UNITED STATES
(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