## Geometry & Topology

### 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.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859285

Digital Object Identifier
doi:10.2140/gt.2017.21.3159

Mathematical Reviews number (MathSciNet)
MR3687116

Zentralblatt MATH identifier
1378.57024

#### Citation

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. https://projecteuclid.org/euclid.gt/1510859285

#### References

• I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045–1087
• G Besson, G Courtois, S Gallot, Inégalités de Milnor–Wood géométriques, Comment. Math. Helv. 82 (2007) 753–803
• R Brooks, W Goldman, The Godbillon–Vey invariant of a transversely homogeneous foliation, Trans. Amer. Math. Soc. 286 (1984) 651–664
• R Brooks, W Goldman, Volumes in Seifert space, Duke Math. J. 51 (1984) 529–545
• P Derbez, Y Liu, S Wang, Chern–Simons theory, surface separability, and volumes of $3$–manifolds, J. Topol. 8 (2015) 933–974
• P Derbez, H B Sun, S C Wang, Finiteness of mapping degree sets for $3$–manifolds, Acta Math. Sin. $($Engl. Ser.$)$ 27 (2011) 807–812
• P Derbez, S Wang, Finiteness of mapping degrees and ${\rm PSL}(2,{\mathbb R})$–volume on graph manifolds, Algebr. Geom. Topol. 9 (2009) 1727–1749
• P Derbez, S Wang, Graph manifolds have virtually positive Seifert volume, J. Lond. Math. Soc. 86 (2012) 17–35
• A Gaifullin, Universal realisators for homology classes, Geom. Topol. 17 (2013) 1745–1772
• M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982) 5–99
• F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551–1620
• W H Jaco, P B Shalen, Seifert fibered spaces in $3$–manifolds, Mem. Amer. Math. Soc. 220, Amer. Math. Soc., Providence, RI (1979)
• K Johannson, Homotopy equivalences of $3$–manifolds with boundaries, Lecture Notes in Math. 761, Springer (1979)
• J Kahn, V Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. 175 (2012) 1127–1190
• R Kirby, Problems in low dimensional manifold theory, from “Algebraic and geometric topology, II” (R J Milgram, editor), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc., Providence, RI (1978) 273–312
• Y Liu, A characterization of virtually embedded subsurfaces in $3$–manifolds, Trans. Amer. Math. Soc. 369 (2017) 1237–1264
• Y Liu, V Markovic, Homology of curves and surfaces in closed hyperbolic $3$–manifolds, Duke Math. J. 164 (2015) 2723–2808
• J Luecke, Y-Q Wu, Relative Euler number and finite covers of graph manifolds, from “Geometric topology” (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 80–103
• W D Neumann, Commensurability and virtual fibration for graph manifolds, Topology 36 (1997) 355–378 \goodbreak
• P Przytycki, D T Wise, Mixed $3$–manifolds are virtually special, preprint (2012)
• P Przytycki, D T Wise, Graph manifolds with boundary are virtually special, J. Topol. 7 (2014) 419–435
• P Przytycki, D T Wise, Separability of embedded surfaces in $3$–manifolds, Compos. Math. 150 (2014) 1623–1630
• A Reznikov, Rationality of secondary classes, J. Differential Geom. 43 (1996) 674–692
• J H Rubinstein, S Wang, $\pi_1$–injective surfaces in graph manifolds, Comment. Math. Helv. 73 (1998) 499–515
• P Scott, The geometries of $3$–manifolds, Bull. London Math. Soc. 15 (1983) 401–487
• T Soma, The Gromov invariant of links, Invent. Math. 64 (1981) 445–454
• H Sun, Virtual domination of $3$–manifolds, Geom. Topol. 19 (2015) 2277–2328
• W P Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m {\unhbox0
• W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982) 357–381
• S C Wang, Y Q Wu, Covering invariants and co-Hopficity of $3$–manifold groups, Proc. London Math. Soc. 68 (1994) 203–224
• D T Wise, From riches to raags: $3$–manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics 117, Amer. Math. Soc., Providence, RI (2012)
• F Yu, S Wang, Covering degrees are determined by graph manifolds involved, Comment. Math. Helv. 74 (1999) 238–247