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Separability of embedded surfaces in 3-manifolds

Part of: Structure and classification of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  27 August 2014

Piotr Przytycki  and
Daniel T. Wise
Piotr Przytycki
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland email [email protected]
Daniel T. Wise
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada H3A 0B9 email [email protected]
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Abstract

We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ is a properly embedded $\pi _1$-injective surface in a compact 3-manifold $M$, then $\pi _1S$ is separable in $\pi _1M$.

Keywords

3-manifoldsurfaceseparable

MSC classification

Secondary: 57M99: None of the above, but in this section 20E26: Residual properties and generalizations; residually finite groups
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 9 , September 2014 , pp. 1623 - 1630
Copyright
© The Author(s) 2014 

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References

Agol, I., The virtual Haken conjecture, with an Appendix by Ian Agol, Daniel Groves, and Jason Manning, Preprint (2012), arXiv:1204.2810.Google Scholar
Bonahon, F., Bouts des variétés hyperboliques de dimension 3, Ann. of Math. (2) 124 (1986), 71–158.Google Scholar
Friedl, S. and Vidussi, S., A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds, J. Eur. Math. Soc. (JEMS) 15 (2013), 2027–2041.Google Scholar
Hempel, J., Residual finiteness for 3-manifolds, Combinatorial group theory and topology (Alta, Utah, 1984), Annals of Mathematics Studies, vol. 111 (Princeton University Press, Princeton, NJ, 1987), 379–396.Google Scholar
Luecke, J. and Wu, Y.-Q., Relative Euler number and finite covers of graph manifolds, in Geometric topology (Athens, GA, 1993), AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 80–103.Google Scholar
Martínez-Pedroza, E., Combination of quasiconvex subgroups of relatively hyperbolic groups, Groups Geom. Dyn. 3 (2009), 317–342.Google Scholar
Przytycki, P. and Wise, D. T., Graph manifolds with boundary are virtually special, J. Topol. 7(2) (2014), 419–435.Google Scholar
Rubinstein, J. H. and Wang, S., π 1-injective surfaces in graph manifolds, Comment. Math. Helv. 73 (1998), 499–515.CrossRefGoogle Scholar
Scott, P., Subgroups of surface groups are almost geometric, J. Lond. Math. Soc. (2) 17 (1978), 555–565.Google Scholar
Silver, D. S. and Williams, S. G., Twisted Alexander polynomials and representation shifts, Bull. Lond. Math. Soc. 41 (2009), 535–540.Google Scholar
Thurston, W. P., The geometry and topology of three-manifolds (1980), Princeton University course notes, available at http://www.msri.org/publications/books/gt3m/.Google Scholar
Wise, D. T., The structure of groups with quasiconvex hierarchy (2011), submitted, available athttp://www.math.mcgill.ca/wise/papers.html.Google Scholar