Endomorphisms of hyperbolic groups. I: The Hopf property.

*(English)*Zbl 0929.20033A 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.

Reviewer: A.Papadopoulos (Strasbourg)

##### MSC:

20F67 | Hyperbolic groups and nonpositively curved groups |

57M07 | Topological methods in group theory |

20E36 | Automorphisms of infinite groups |