Skip to main content
SpringerLink
Log in

Virtual specialness of certain graphs of special cube complexes

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We investigate the virtual specialness of a compact cube complex X that splits as a graph of nonpositively curved cube complexes. We prove virtual specialness of X when each vertex space of X has word-hyperbolic \(\pi _1\) and \(\pi _1X\) has “finite stature” relative to its edge groups. The results generalize the motivating case when tree \(\times \) tree lattices are virtual products.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Burger, M., Mozes, S.: Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92, 151–194 (2001)

    Article  MathSciNet  Google Scholar 

  2. Gitik, R.: Doubles of groups and hyperbolic LERF 3-manifolds. Ann. Math. (2) 150(3), 775–806 (1999)

    Article  MathSciNet  Google Scholar 

  3. Gitik, R., Mitra, M., Rips, E., Sageev, M.: Widths of subgroups. Trans. Am. Math. Soc. 350(1), 321–329 (1998)

    Article  MathSciNet  Google Scholar 

  4. Haglund, F.: Commensurability and separability of quasiconvex subgroups. Algebr. Geom. Topol. 6, 949–1024 (2006) (electronic)

  5. Haglund, F., Wise, D.T.: Special cube complexes. Geom. Funct. Anal. 17(5), 1551–1620 (2008)

    Article  MathSciNet  Google Scholar 

  6. Haglund, F., Wise, D.T.: Coxeter groups are virtually special. Adv. Math. 224(5), 1890–1903 (2010)

    Article  MathSciNet  Google Scholar 

  7. Haglund, F., Wise, D.T.: A combination theorem for special cube complexes. Ann. Math. (2) 176(3), 1427–1482 (2012)

    Article  MathSciNet  Google Scholar 

  8. Hall, M., Jr.: Coset representations in free groups. Trans. Am. Math. Soc. 67, 421–432 (1949)

    Article  MathSciNet  Google Scholar 

  9. Hruska, G.C., Wise, D.T.: Packing subgroups in relatively hyperbolic groups. Geom. Topol. 13(4), 1945–1988 (2009)

    Article  MathSciNet  Google Scholar 

  10. Huang, J., Kleiner, B.: Groups quasi-isometric to right-angled Artin groups. Duke Math. J. 167(3), 537–602 (2018)

    Article  MathSciNet  Google Scholar 

  11. Huang, J.: Commensurability of groups quasi-isometric to RAAGs. Invent. Math. 213(3), 1179–1247 (2018)

    Article  MathSciNet  Google Scholar 

  12. Huang, J., Wise, D.T.: Stature and separability in graphs of groups, pp. 1–33. arXiv:1904.06021 (2019)

  13. Kapovich, I., Short, H.: Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups. Can. J. Math. 48(6), 1224–1244 (1996)

    Article  MathSciNet  Google Scholar 

  14. Liu, Y.: Virtual cubulation of nonpositively curved graph manifolds. J. Topol. 6(4), 793–822 (2013)

    Article  MathSciNet  Google Scholar 

  15. Minasyan, A.: Separable subsets of GFERF negatively curved groups. J. Algebra 304(2), 1090–1100 (2006)

    Article  MathSciNet  Google Scholar 

  16. Przytycki, P., Wise, D.T.: Graph manifolds with boundary are virtually special. J. Topol. 7(2), 419–435 (2014)

    Article  MathSciNet  Google Scholar 

  17. Przytycki, P., Wise, D.T.: Mixed 3-manifolds are virtually special. J. Am. Math. Soc. 31(2), 319–347 (2018)

    Article  MathSciNet  Google Scholar 

  18. Ribes, L., Zalesskii, P.A.: On the profinite topology on a free group. Bull. Lond. Math. Soc. 25(1), 37–43 (1993)

    Article  MathSciNet  Google Scholar 

  19. Wise, D.T.: Non-positively curved squared complexes, aperiodic tilings, and non-residually finite groups. PhD thesis, Princeton University (1996)

  20. Wise, D.T.: Subgroup separability of the figure 8 knot group. Topology 45(3), 421–463 (2006)

    Article  MathSciNet  Google Scholar 

  21. Wise, D.T.: The structure of groups with a quasiconvex hierarchy. In: The Structure of Groups with a Quasiconvex Hierarchy. Princeton University Press, Princeton (2021)

Download references

Author information

Author notes

  1. Jingyin Huang

    Present address: Department of Mathematics, The Ohio State University, 231 W. 18th Ave, Columbus, OH, 43210, USA

Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada

    Jingyin Huang & Daniel T. Wise

Authors
  1. Jingyin Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel T. Wise
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jingyin Huang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, J., Wise, D.T. Virtual specialness of certain graphs of special cube complexes. Math. Ann. 388, 329–357 (2024). https://doi.org/10.1007/s00208-022-02527-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-022-02527-0

Search

Navigation