Weil–Petersson translation length and manifolds with many fibered fillings

Abstract We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil–Petersson translation length contains a finite set of transverse and level closed curves with the property that drilling out this set of curves results in one of a finite number of cusped hyperbolic 3–manifolds. Moreover, the set of resulting manifolds depends only on the bound for normalized translation length. This gives a Weil–Petersson analog of a theorem of Farb–Leininger–Margalit [9] about Teichmuller translation length. We also prove a complementary result that explains the necessity of removing level curves by producing new estimates for the Weil–Petersson translation length of compositions of pseudo-Anosov mapping classes and arbitrary powers of a Dehn twist.