Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T09:18:46.341Z Has data issue: false hasContentIssue false
  • English
  • Français

Virtual 1-domination of 3-manifolds

Part of: Low-dimensional topology

Published online by Cambridge University Press:  27 February 2018

Yi Liu  and
Hongbin Sun
Yi Liu
Affiliation:
Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China email liuyi@math.pku.edu.cn
Hongbin Sun
Affiliation:
Department of Mathematics, UC Berkeley, CA 94720, USA email hongbin.sun@rutgers.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

good pantsnon-zero degree mapvirtual property

MSC classification

Primary: 57M10: Covering spaces 57M50: Geometric structures on low-dimensional manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 3 , March 2018 , pp. 621 - 639
Copyright
© The Authors 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agol, I., The virtual Haken conjecture, with an appendix by I. Agol, D. Groves, J. Manning , Doc. Math. 18 (2013), 1045–1087.Google Scholar
Boileau, M. and Wang, S., Non-zero degree maps and surface bundles over S 1 , J. Differential Geom. 43 (1996), 789–806.Google Scholar
Hamenstädt, U., Incompressible surfaces in rank one locally symmetric spaces , Geom. Funct. Anal. 25 (2015), 815–859.CrossRefGoogle Scholar
Kahn, J. and Markovic, V., Immersing almost geodesic surfaces in a closed hyperbolic three manifold , Ann. of Math. (2) 175 (2012), 1127–1190..Google Scholar
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].Google Scholar
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].Google Scholar
Liu, Y. and Markovic, V., Homology of curves and surfaces in closed hyperbolic 3-manifolds , Duke Math. J. 164 (2015), 2723–2808.Google Scholar
Šarić, D., Complex Fenchel–Nielsen coordinates with small imaginary parts , Trans. Amer. Math. Soc. 366 (2014), 6541–6565.Google Scholar
Scott, P., Subgroups of surface groups are almost geometric , J. Lond. Math. Soc. (2) 17 (1978), 555–565.Google Scholar
Sun, H., Virtual homological torsion of closed hyperbolic 3-manifolds , J. Differential Geom. 100 (2015), 547–583.Google Scholar
Sun, H., Virtual domination of 3-manifolds , Geom. Topol. 19 (2015), 2277–2328.Google Scholar
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).Google Scholar