Skip to main content
SpringerLink
Log in

Geometry of right-angled Coxeter groups on the Croke–Kleiner spaces

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

In this paper we study the right-angled Coxeter groups that acts geometrically on the Salvetti complex of a certain right-angled Artin group, which we refer to as Croke–Kleiner spaces. We prove that any right-angled Coxeter group that acts geometrically on the Croke–Kleiner spaces acts with \(\pi /2\) angles between reflecting axes, while the quasi-isometric right-angled Artin group can act with angles that are any real number in the range \((0, \pi /2]\). The contrast between the two examples shows that in this case a right-angled Coxeter group is geometrically more “rigid” than its quasi-isometric counterpart.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)

    Google Scholar 

  2. Croke, C.B., Kleiner, B.: Spaces with nonpositive curvature and their ideal boundaries. Topology 39(3), 549–556 (2000). doi:10.1016/S0040-9383(99)00016-6

    Article  MathSciNet  MATH  Google Scholar 

  3. Davis, M.W., Januszkiewicz, T.: Right-angled Artin groups are commensurable with right-angled Coxeter groups. J. Pure Appl. Algebra 153(3), 229–235 (2000). doi:10.1016/S0022-4049(99)00175-9

    Article  MathSciNet  MATH  Google Scholar 

  4. Gromov, M.: Groups of polynomial growth and expanding maps. Inst Hautes Études Sci Publ Math (53), 53–73, (1981) http://www.numdam.org/item?id=PMIHES_1981__53__53_0

  5. Gutierrez, M., Piggott, A.: Rigidity of graph products of abelian groups. Bull. Aust. Math. Soc. 77(2), 187–196 (2008). doi:10.1017/S0004972708000105

    Article  MathSciNet  MATH  Google Scholar 

  6. Mihalik, M., Ruane, K., Tschantz, S.: Local connectivity of right-angled Coxeter group boundaries. J. Group Theory 10(4), 531–560 (2007). doi:10.1515/JGT.2007.042

    Article  MathSciNet  MATH  Google Scholar 

  7. Moussong, G.: Hyperbolic Coxeter groups. Ph.D. thesis, The Ohio State University, Columbus, Ohio (1988)

  8. Serre, J.P.: Trees. Springer, Berlin (1980). (translated from the French by John Stillwell)

    Book  MATH  Google Scholar 

  9. Wilson, J.: A CAT(0) group with uncountably many distinct boundaries. J. Group Theory 8(2), 229–238 (2005). doi:10.1515/jgth.2005.8.2.229

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Yulan Qing

Authors
  1. Yulan Qing
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yulan Qing.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qing, Y. Geometry of right-angled Coxeter groups on the Croke–Kleiner spaces. Geom Dedicata 183, 113–122 (2016). https://doi.org/10.1007/s10711-016-0149-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-016-0149-1

Keywords

Mathematics Subject Classification

Search

Navigation