## Osaka Journal of Mathematics

### The virtual Thurston seminorm of 3-manifolds

#### Abstract

We show that the Thurston seminorms of all finite covers of an aspherical 3-manifold determine whether it is a graph manifold, a mixed 3-manifold or hyperbolic.

#### Article information

Source
Osaka J. Math., Volume 56, Number 1 (2019), 51-63.

Dates
First available in Project Euclid: 16 January 2019

https://projecteuclid.org/euclid.ojm/1547607625

Mathematical Reviews number (MathSciNet)
MR3908776

Zentralblatt MATH identifier
07055398

#### Citation

Boileau, Michel; Friedl, Stefan. The virtual Thurston seminorm of 3-manifolds. Osaka J. Math. 56 (2019), no. 1, 51--63. https://projecteuclid.org/euclid.ojm/1547607625

#### References

• I. Agol: Criteria for virtual fibering, J. Topol. 1 (2008), 269–284.
• I. Agol: The virtual Haken conjecture, with an appendix by I. Agol, D. Groves and J. Manning, Doc. Math. 18 (2013), 1045–1087.
• M. Aschenbrenner and S. Friedl: $3$-manifold groups are virtually residually $p$, Mem. Amer. Math. Soc. 225 (2013), no. 1058.
• M. Aschenbrenner, S. Friedl and H. Wilton: 3-manifold groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zurich, 2015.
• M. Boileau and S. Friedl: The profinite completion of $3$-manifold groups, fiberedness and the Thurston norm, arXiv:1505.07799.
• M.R. Bridson and A.W. Reid: Profinite rigidity, fibering, and the figure-eight knot, arXiv:1505.07886.
• J. Deblois, S. Friedl and S. Vidussi: The rank gradient for infinite cyclic covers of 3-manifolds, Michigan Math. J. 63 (2014), 65–81.
• D. Eisenbud and W. Neumann: Three-dimensional Link Theory and Invariants of Plane Curve Singularities, Annals of Mathematics Studies, 110, Princeton University Press, Princeton, NJ, 1985.
• S. Friedl and T. Kitayama: The virtual fibering theorem for $3$-manifolds, Enseign. Math. 60 (2014), 79–107.
• S. Friedl and S. Vidussi: Symplectic $S^{1} \times N^3$, surface subgroup separability, and vanishing Thurston norm, J. Amer. Math. Soc. 21 (2008), 597–610.
• D. Cooper and D. Futer: Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds, Preprint (2017).
• D. Gabai: Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983), 445–503.
• D. Groves and J. Manning: Quasiconvexity and Dehn filling, Preprint (2017).
• J. Hempel: $3$-Manifolds, Ann. of Math. Studies, 86, Princeton University Press, Princeton, NJ, 1976.
• J. Hempel: Residual finiteness for $3$-manifolds; in Combinatorial group theory and topology (Alta, Utah, 1984), Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987, 379–396.
• T. Le: Growth of homology torsion in finite coverings and hyperbolic volume, preprint (2014), arXiv:1412.7758, to appear in the Annales de l'Institut Fourier.
• D. Long and A. Reid: Surface subgroups and subgroup separability in $3$-manifold topology, Publicações Matemáticas do IMPA, Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, 2005.
• P. Przytycki and D. Wise: Mixed $3$-manifolds are virtually special, J. Amer. Math. Soc. 31 (2018), 319–347.
• P. Przytycki and D. Wise: Separability of embedded surfaces in $3$-manifolds, Compos. Math. 150 (2014), 1623–1630.
• W. Thurston: A norm for the homology of $3$-manifolds, Mem. Amer. Math. Soc. 59 (1986), 99–130.
• G. Wilkes: Profinite rigidity of graph manifolds and JSJ decompositions of 3-manifolds, preprint (2016), arXiv:1605.08244.
• H. Wilton and P. Zalesskii: Profinite properties of graph manifolds, Geom. Dedicata 147 (2010), 29–45.
• H. Wilton and P. Zalesskii: Distinguishing geometries using finite quotients, Geom. Topol. 21 (2017), 345–384.
• D. Wise: The structure of groups with a quasi-convex hierarchy, Electron Res. Announc. Math. Sci 16 (2009), 44–55.
• D. Wise: The structure of groups with a quasi-convex hierarchy, preprint (2012), http://www.math.mcgill.ca/wise/papers.html (accessed October 29, 2012).
• D. Wise; From riches to RAAGs: $3$-manifolds, right–angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics 117, 2012.