Virtual domination of 3–manifolds

Sun, Hongbin¹

¹ Department of Mathematics, University of California, Berkeley, 751 Evans Hall, Berkeley, CA 94720, USA

Geometry & topology, Tome 19 (2015) no. 4, pp. 2277-2328.

Résumé

For any closed oriented hyperbolic $\mathfrak{3}$–manifold $M$ and any closed oriented $\mathfrak{3}$–manifold $N$, we show that $M$ admits a finite cover $M^{\prime }$ such that there exists a degree-$\mathfrak{2}$ map $f:M^{\prime }\to N$, ie $M$ virtually $\mathfrak{2}$–dominates $N$.

Détail

DOI : 10.2140/gt.2015.19.2277

Classification : 57M10, 30F40, 57M50
Keywords: hyperbolic $3$–manifold, finite cover, degree-nonzero map

Bibliographie

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

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

[3] P Derbez, Y Liu, S Wang, Chern–Simons theory, surface separability, and volumes of $3$–manifolds,

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

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

[6] H M Hilden, M T Lozano, J M Montesinos, W C Whitten, On universal groups and three-manifolds, Invent. Math. 87 (1987) 441

[7] J Kahn, V Markovic, The good pants homology and a proof of the Ehrenpreis conjecture,

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

[9] Y Liu, V Markovic, Homology of curves and surfaces in closed hyperbolic $3$–manifolds,

[10] C C Moore, Exponential decay of correlation coefficients for geodesic flows, from: "Group representations, ergodic theory, operator algebras, and mathematical physics" (editor C C Moore), Math. Sci. Res. Inst. Publ. 6, Springer (1987) 163

[11] M Pollicott, Exponential mixing for the geodesic flow on hyperbolic three-manifolds, J. Statist. Phys. 67 (1992) 667

[12] D Šarić, Complex Fenchel–Nielsen coordinates with small imaginary parts, Trans. Amer. Math. Soc. 366 (2014) 6541

[13] P Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. 17 (1978) 555

[14] H Sun, Virtual homological torsion of closed hyperbolic $3$–manifolds,

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

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

[17] S Wang, Non-zero degree maps between $3$–manifolds, from: "Proceedings of the International Congress of Mathematicians, Vol. II" (editor T Li), Higher Ed. Press, Beijing (2002) 457

[18] D Wise, The structure of groups with a quasiconvex hierarchy, (2011)