Generalized Hopfian property, minimal Haken manifold, and J. Simon's   conjecture for 3-manifold groups
  We address a conjecture that $\pi_1$-surjective maps between closed aspherical 3-manifolds having the same rank on $\pi_1$ must be of non-zero degree. The conjecture is proved for Seifert manifolds, which is used in constructing the first known example of minimum Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3-manifolds. 
Alan W. Reid, Shicheng Wang, Qing Zhou