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

Derbez, Pierre¹ ; Liu, Yi² ; Sun, Hongbin³ ; Wang, Shicheng⁴

¹ Aix Marseille Université, CNRS UMR 7373, Centrale Marseille, I2M, Marseille, France
² Beijing International Center for Mathematical Research, Peking University, Beijing, China
³ Department of Mathematics, University of California, Berkeley, CA, United States, Department of Mathematics, Rutgers University, Piscataway, NJ, United States
⁴ School of Mathematical Sciences, Peking University, Beijing, China

Geometry & topology, Tome 21 (2017) no. 5, pp. 3159-3190.

Résumé

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.

Détail

DOI : 10.2140/gt.2017.21.3159

Classification : 57M50, 51H20
Keywords: Seifert volume, nonzero degree maps, growth rate

Bibliographie

[1] I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045

[2] G Besson, G Courtois, S Gallot, Inégalités de Milnor–Wood géométriques, Comment. Math. Helv. 82 (2007) 753 | DOI

[3] R Brooks, W Goldman, The Godbillon–Vey invariant of a transversely homogeneous foliation, Trans. Amer. Math. Soc. 286 (1984) 651 | DOI

[4] R Brooks, W Goldman, Volumes in Seifert space, Duke Math. J. 51 (1984) 529 | DOI

[5] P Derbez, Y Liu, S Wang, Chern–Simons theory, surface separability, and volumes of 3–manifolds, J. Topol. 8 (2015) 933 | DOI

[6] P Derbez, H B Sun, S C Wang, Finiteness of mapping degree sets for 3–manifolds, Acta Math. Sin. Engl. Ser. 27 (2011) 807 | DOI

[7] P Derbez, S Wang, Finiteness of mapping degrees and PSL(2, R)–volume on graph manifolds, Algebr. Geom. Topol. 9 (2009) 1727 | DOI

[8] P Derbez, S Wang, Graph manifolds have virtually positive Seifert volume, J. Lond. Math. Soc. 86 (2012) 17 | DOI

[9] A Gaifullin, Universal realisators for homology classes, Geom. Topol. 17 (2013) 1745 | DOI

[10] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982) 5

[11] F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551 | DOI

[12] W H Jaco, P B Shalen, Seifert fibered spaces in 3–manifolds, 220, Amer. Math. Soc. (1979) | DOI

[13] K Johannson, Homotopy equivalences of 3–manifolds with boundaries, 761, Springer (1979) | DOI

[14] J Kahn, V Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. 175 (2012) 1127 | DOI

[15] R Kirby, Problems in low dimensional manifold theory, from: "Algebraic and geometric topology, II" (editor R J Milgram), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 273

[16] Y Liu, A characterization of virtually embedded subsurfaces in 3–manifolds, Trans. Amer. Math. Soc. 369 (2017) 1237 | DOI

[17] Y Liu, V Markovic, Homology of curves and surfaces in closed hyperbolic 3–manifolds, Duke Math. J. 164 (2015) 2723 | DOI

[18] J Luecke, Y Q Wu, Relative Euler number and finite covers of graph manifolds, from: "Geometric topology" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 80

[19] W D Neumann, Commensurability and virtual fibration for graph manifolds, Topology 36 (1997) 355 | DOI

[20] P Przytycki, D T Wise, Mixed 3–manifolds are virtually special, preprint (2012)

[21] P Przytycki, D T Wise, Graph manifolds with boundary are virtually special, J. Topol. 7 (2014) 419 | DOI

[22] P Przytycki, D T Wise, Separability of embedded surfaces in 3–manifolds, Compos. Math. 150 (2014) 1623 | DOI

[23] A Reznikov, Rationality of secondary classes, J. Differential Geom. 43 (1996) 674 | DOI

[24] J H Rubinstein, S Wang, π1–injective surfaces in graph manifolds, Comment. Math. Helv. 73 (1998) 499 | DOI

[25] P Scott, The geometries of 3–manifolds, Bull. London Math. Soc. 15 (1983) 401 | DOI

[26] T Soma, The Gromov invariant of links, Invent. Math. 64 (1981) 445 | DOI

[27] H Sun, Virtual domination of 3–manifolds, Geom. Topol. 19 (2015) 2277 | DOI

[28] W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)

[29] W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982) 357 | DOI

[30] S C Wang, Y Q Wu, Covering invariants and co-Hopficity of 3–manifold groups, Proc. London Math. Soc. 68 (1994) 203 | DOI

[31] D T Wise, From riches to raags : 3–manifolds, right-angled Artin groups, and cubical geometry, 117, Amer. Math. Soc. (2012) | DOI

[32] F Yu, S Wang, Covering degrees are determined by graph manifolds involved, Comment. Math. Helv. 74 (1999) 238 | DOI