  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 presisely 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 Poincare-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.
