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

Pierre Derbez, Yi Liu, Hongbin Sun and Shicheng Wang

Geometry & Topology 21 (2017) 3159–3190

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.

Seifert volume, nonzero degree maps, growth rate
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 51H20
Received: 6 July 2016
Accepted: 23 December 2016
Published: 15 August 2017
Proposed: Cameron Gordon
Seconded: Gang Tian, Walter Neumann
Pierre Derbez
Aix Marseille Université
CNRS UMR 7373, Centrale Marseille, I2M
Yi Liu
Beijing International Center for Mathematical Research
Peking University
Hongbin Sun
Department of Mathematics
University of California
Berkeley, CA
United States
Department of Mathematics
Rutgers University
Piscataway, NJ
United States
Shicheng Wang
School of Mathematical Sciences
Peking University