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Profinite rigidity of graph manifolds, II: knots and mapping classes

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Abstract

In this paper we study some consequences of the author’s classification of graph manifolds by their profinite fundamental groups. In particular we study commensurability, the behaviour of knots, and relation to mapping classes. We prove that the exteriors of graph knots are distinguished among all 3-manifold groups by their profinite fundamental groups. We also prove a strong conjugacy separability result for certain mapping classes of surfaces.

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References

  1. M. Boileau and S. Friedl, Profinite completions and 3-manifold groups, survey article, https://doi.org/www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/04Boileau-Friedl_ver3.pdf.

  2. M. Boileau and S. Friedl, The profinite completion of 3-manifold groups, fiberedness and the Thurston norm, preprint, https://doi.org/abs/1505.07799.

  3. M. R. Bridson, A. W. Reid and H. Wilton, Profinite rigidity and surface bundles over the circle, Bulletin of the London Mathematical Society 49 (2017), 831–841.

    Article  MathSciNet  Google Scholar 

  4. R. Budney, JSJ-decompositions of knot and link complements in the S 3, L’enseignement Mathématique 52 (2006), 319–359.

    MathSciNet  MATH  Google Scholar 

  5. D. Eisenbud and W. D. Neumann, Three-dimensional Link Theory and Invariants of Plane Curve Singularities, Annals of Mathematics Studies, Vol. 110, Princeton University Press, Princeton, NJ, 1985.

  6. C. McA. Gordon and J. Luecke, Knots are determined by their complements, Journal of the American Mathematical Society 2 (1989), 371–415.

    Article  MathSciNet  Google Scholar 

  7. F. Grunewald and P. Zalesskii, Genus for groups, Journal of Algebra 326 (2011), 130–168.

    Article  MathSciNet  Google Scholar 

  8. J. Hempel, Some 3-manifold groups with the same finite quotients, preprint, https://doi.org/abs/1409.3509.

  9. A. Jaikin-Zapirain, Recognition of being fibred for compact 3-manifolds, preprint, https://doi.org/verso.mat.uam.es/~andrei.jaikin/preprints/fibering.pdf.

  10. P. Orlik, Seifert Manifolds, Lecture Notes in Mathematics, Vol. 291, Springer, Berlin-New York, 2006.

  11. L. Ribes and P. Zalesskii, Profinite Groups, Ergebnisse der Mathematik und ihrer Gren-zgebiete, Vol. 40, Springer, Berlin, 2000.

  12. J.-P. Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.

    Google Scholar 

  13. J. Stallings, On fibering certain 3-manifolds, in Topology of 3-manifolds and Related Topics, Proceedings of the University of Georgia Institute, Prentice Hall, Englewood Cliffs, NJ, 1962, pp. 95–100.

    MATH  Google Scholar 

  14. G. Wilkes, Profinite rigidity for Seifert fibre spaces, Geometriae Dedicata 188 (2017), 141–163.

    Article  MathSciNet  Google Scholar 

  15. G. Wilkes, Profinite rigidity of graph manifolds and JSJ decompositions of 3-manifolds, Journal of Algebra 502 (2018), 538–587.

    Article  MathSciNet  Google Scholar 

  16. H. Wilton and P. Zalesskii, Profinite properties of graph manifolds, Geometriae Dedicata 147 (2010), 29–45.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Gareth Wilkes

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  1. Gareth Wilkes
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Correspondence to Gareth Wilkes.

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Wilkes, G. Profinite rigidity of graph manifolds, II: knots and mapping classes. Isr. J. Math. 233, 351–378 (2019). https://doi.org/10.1007/s11856-019-1908-0

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  • DOI: https://doi.org/10.1007/s11856-019-1908-0

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