Geometry & Topology

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

Pierre Derbez, Yi Liu, Hongbin Sun, and Shicheng Wang

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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.

Article information

Geom. Topol., Volume 21, Number 5 (2017), 3159-3190.

Received: 6 July 2016
Accepted: 23 December 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 51H20: Topological geometries on manifolds [See also 57-XX]

Seifert volume nonzero degree maps growth rate


Derbez, Pierre; Liu, Yi; Sun, Hongbin; Wang, Shicheng. Positive simplicial volume implies virtually positive Seifert volume for $3$–manifolds. Geom. Topol. 21 (2017), no. 5, 3159--3190. doi:10.2140/gt.2017.21.3159.

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