Nonzero degree maps between closed orientable three-manifolds

Abstract

This paper adresses the following problem: Given a closed orientable three-manifold $M$, are there at most finitely many closed orientable three-manifolds 1-dominated by $M$? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincaré-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold $M$ 1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of $M$.