A group G is called hopfian if each self-epimorphism of G is an isomorphism. A group G is called cohopfian if each self-monomorphism of G is an isomorphism. The authors call the fundamental groups of 3-manifolds the groups of 3-manifolds. A compact, connected, orientable 3-manifold M is called geometric (in the sense of Thurston), if M is either a Seifert manifold, or a hyperbolic manifold, or a Haken manifold, or a connected sum of such manifolds. A famous conjecture is that all compact orientable 3-manifolds are geometric. Following a theorem of Thurston, the groups of geometric 3-manifolds are hopfian. The aim of the paper is to give a complete description when the groups of geometric 3-manifolds as well as the finite generated torsion free Kleinian groups are cohopfian.
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