Epimorphisms of 3-manifold groups

Abstract

Let $f\colon M\to N$ be a proper map between two aspherical compact orientable 3-manifolds with empty or toroidal boundary. We assume that $N$ is not a closed graph-manifold. Suppose that $f$ induces an epimorphism on fundamental groups. We show that $f$ is homotopic to a homeomorphism if one of the following holds: either for any finite-index subgroup $\Gamma$ of $\pi_1(N)$ the ranks of $\Gamma$ and of $f_*^{-1}(\Gamma)$ agree, or for any finite cover $\tilde{N}$ of $N$ the Heegaard genus of $\tilde{N}$ and the Heegaard genus of the pull-back cover $\tilde{M}$ agree.