Positive simplicial volume implies virtually positive Seifert volume for $3$–manifolds

Abstract

We show that for any closed orientable $3$–manifold with positive simplicial volume, the growth of the Seifert volume of its finite covers is faster than the linear rate. In particular, each closed orientable $3$–manifold with positive simplicial volume has virtually positive Seifert volume. The result reveals certain fundamental differences between the representation volumes of hyperbolic type and Seifert type. The proof is based on developments and interactions of recent results on virtual domination and on virtual representation volumes of $3$–manifolds.