Endomorphisms of hyperbolic groups. I: The Hopf property
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.
scientific article; zbMATH DE number 1307828