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

TL;DR

This paper demonstrates that closed orientable 3-manifolds with positive simplicial volume have finite covers with Seifert volume growing faster than linearly, indicating a fundamental difference between hyperbolic and Seifert representation volumes.

Contribution

It establishes that positive simplicial volume implies virtually positive Seifert volume, revealing new insights into the growth behavior of representation volumes in 3-manifolds.

Findings

Finite covers of such manifolds have superlinear Seifert volume growth

Positive simplicial volume implies virtually positive Seifert volume

Highlights differences between hyperbolic and Seifert volume types

Abstract

In this paper, it is shown 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 difference between the representation volume of the hyperbolic type and the Seifert type. The proof is based on developments and reactions of recent results on virtual domination and on virtual representation volumes of $3$-manifolds.