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Endomorphisms of hyperbolic groups. I: The Hopf property. (English) Zbl 0929.20033
A group is said to have the Hopf property if every surjective endomorphism of this group is injective. The author proves that a torsion-free hyperbolic group has the Hopf property. The main tool used in the proof is the canonical (JSJ) decomposition of a hyperbolic group which the author has developed in a series of previous papers (in part with E. Rips), where he established in this abstract group-theoretic setting the theory of Jaco-Shalen and Johannson of the canonical decomposition of a 3-manifold by using the characteristic submanifold. As a corollary, the author proves that if $$M$$ and $$N$$ are closed negatively curved manifolds and if there exist degree 1 maps from $$M$$ to $$N$$ and from $$N$$ to $$M$$, then $$M$$ is homotopy equivalent to $$N$$. It is known then that this implies, if $$\dim(M)\geq 5$$, that $$M$$ and $$N$$ are homeomorphic.

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 57M07 Topological methods in group theory 20E36 Automorphisms of infinite groups
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