ENDOMORPHISMS OF HYPERBOLIC GROUPS I: THE HOPF PROPERTY

Using the canonical JSJ decomposition [9,12] we prove that every torsion-free hyperbolic group is Hopf. This implies, in particular, that if M and N are closed negatively curved manifolds and there exist degree 1 maps from M to N and from N to M, then M is homotopy equivalent to N. By [5] if in addition dim(M)⩾5 then M and N are homeomorphic.