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Virtual 1-domination of 3-manifolds

  • Yi Liu (a1) and Hongbin Sun (a2)


It is shown in this paper that given any closed oriented hyperbolic 3-manifold, every closed oriented 3-manifold is mapped onto by a finite cover of that manifold via a map of degree 1, or in other words, virtually 1-dominated by that manifold. This improves a known result of virtual 2-domination. The proof invokes a recently developed enhanced version of the connection principle in good pants constructions.



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[Agol13] Agol, I., The virtual Haken conjecture, with an appendix by I. Agol, D. Groves, J. Manning , Doc. Math. 18 (2013), 10451087.
[BW96] Boileau, M. and Wang, S., Non-zero degree maps and surface bundles over S 1 , J. Differential Geom. 43 (1996), 789806.
[Ham15] Hamenstädt, U., Incompressible surfaces in rank one locally symmetric spaces , Geom. Funct. Anal. 25 (2015), 815859.
[KM12] Kahn, J. and Markovic, V., Immersing almost geodesic surfaces in a closed hyperbolic three manifold , Ann. of Math. (2) 175 (2012), 11271190..
[LL16] Lakeland, G. and Leininger, C., Strict contractions and exotic $\text{SO}_{0}(d;1)$ quotients, J. Lond. Math. Soc., to appear. Preprint (2016), arXiv:1606.02609v4 [math.GT].
[Liu16] Liu, Y., Immersing quasi-Fuchsian surfaces of odd Euler characteristic in closed hyperbolic 3-manifolds, J. Diff. Geom., to appear. Preprint (2016), arXiv:1603.07069v2 [math.GT].
[LM15] Liu, Y. and Markovic, V., Homology of curves and surfaces in closed hyperbolic 3-manifolds , Duke Math. J. 164 (2015), 27232808.
[Sar14] Šarić, D., Complex Fenchel–Nielsen coordinates with small imaginary parts , Trans. Amer. Math. Soc. 366 (2014), 65416565.
[Sco78] Scott, P., Subgroups of surface groups are almost geometric , J. Lond. Math. Soc. (2) 17 (1978), 555565.
[Sun15a] Sun, H., Virtual homological torsion of closed hyperbolic 3-manifolds , J. Differential Geom. 100 (2015), 547583.
[Sun15b] Sun, H., Virtual domination of 3-manifolds , Geom. Topol. 19 (2015), 22772328.
[Wise12] Wise, D. T., From riches to raags: 3-manifolds, right-angled artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics, vol. 117 (American Mathematical Society, Providence, RI, 2012).
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