Cofinitely Hopfian groups, open mappings and knot complements

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.

14 pages

References in corpus (6)

• Introduction to Arithmetic Groups
• Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams
• Limit groups and groups acting freely on R^n-trees
• Simplicial volume of closed locally symmetric spaces of non-compact type
• Translation lengths in Out(F_n)
• Mapping tori of free group automorphisms are coherent

Cited by in corpus (4)

• Rigidity of high dimensional graph manifolds
• Explicit bounds for fixed subgroups of endomorphisms of free products
• Volume vs. Complexity of Hyperbolic Groups
• Endomorphisms of mapping tori