Skip to main content
SpringerLink
Log in

Dynamics on the Morse boundary

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

Abstract

Let X be a proper geodesic metric space and let G be a group of isometries of X which acts geometrically. Cordes constructed the Morse boundary of X which generalizes the contracting boundary for CAT(0) spaces and the visual boundary for hyperbolic spaces. We characterize Morse elements in G by their fixed points on the Morse boundary \(\partial _MX\). The dynamics on the Morse boundary is very similar to that of a \(\delta \)-hyperbolic space. In particular, we show that the action of G on \(\partial _MX\) is minimal if G is not virtually cyclic. We also get a uniform convergence result on the Morse boundary which gives us a weak north-south dynamics for a Morse isometry. This generalizes the work of Murray in the case of the contracting boundary of a CAT(0) space.

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

Similar content being viewed by others

References

  1. Alonso J., Uan M.: Thomas Brady, Daryl Cooper, Vincent Ferlini, Martin Lustig, Michael Mihalik, Michael Shapiro, and Hamish Short. Notes on word hyperbolic groups. In Group theory from a geometrical viewpoint (1991)

  2. Antolin, Y., Mj, M., Sisto, A., Taylor, S.J.: Intersection properties of stable subgroups and bounded cohomology. Ind. Univ. Math. J. 68(1), 179–199 (2019)

    Article  MathSciNet  Google Scholar 

  3. Martin, R., Bridson., André, Haefliger.: Metric spaces of non-positive curvature, 319. Springer Science and Business Media (2013)

  4. Benakli, N., Kapovich, I.: Boundaries of hyperbolic groups Combinatorial and geometric group theory (R.Gilman, et al, editors) Contemporary Mathematics 296, 39–94 (2002)

  5. Charney, R., Cordes, M., Murray, D.: Quasi-mobius homeomorphisms of Morse boundaries. Bull. Lond. Math. Soc. 51(3), 501–515 (2019)

    Article  MathSciNet  Google Scholar 

  6. Charney, R., Cordes, M., Sisto, A.: Complete topological descriptions of certain Morse boundaries arXiv preprint arXiv:1908.03542 (2019)

  7. Matthew, C., Durham, M.G.: Boundary convex cocompactness and stability of subgroups of finitely generated groups. Int. Math. Res. Not. IMRN 6, 1699–1724 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Cordes, Matthew, Hume, David: Stability and the Morse boundary. J. Lond. Math. Soci. 95(3), 963–988 (2017)

    Article  MathSciNet  Google Scholar 

  9. Croke, C.B., Kleiner, B.: Spaces with nonpositive curvature and their ideal boundaries. Topology 39(3), 549–556 (2000)

    Article  MathSciNet  Google Scholar 

  10. Cashen, Christopher, Mackay, John: A metrizable topology on the contracting boundary of a group. Trans. Am. Math. Soci. 372(3), 1555–1600 (2019)

    Article  MathSciNet  Google Scholar 

  11. Cordes, Matthew: Morse boundaries of proper geodesic metric spaces. Gr. Geom. Dyn. 11(4), 1281–1306 (2017)

    Article  MathSciNet  Google Scholar 

  12. Charney, Ruth, Sultan, Harold: Contracting boundaries of CAT (0) spaces. J. Topol. 8(1), 93–117 (2014)

    Article  MathSciNet  Google Scholar 

  13. Durham, M.G., Taylor, S.J.: Convex cocompactness and stability in mapping class groups algebraic and geometric. Topology 15(5), 2839–2859 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Fink, E.: Morse geodesics in torsion groups. arXiv preprint arXiv:1710.11191 (2017)

  15. Ghys, Étienne.: Sur les groupes hyperboliques d’apres Mikhael Gromov. Progr. Math. 83 (1990)

  16. Mikhael, G.: Hyperbolic Groups. Essays in Group Theory. Springer, Hoboken (1987)

    Google Scholar 

  17. Hamenstädt, Ursula: Rank-one isometries of proper CAT (0)-spaces. Contemp. Math. 501, 43 (2009)

    Article  MathSciNet  Google Scholar 

  18. Liu, Q.: Maps on the Morse boundary (2020)

  19. Incerti-Medici, M.: Comparing topologies on the Morse boundary and quasi-isometry invariance, arXiv preprint arXiv:1903.07048 (2019)

  20. Morse, H.M.: A fundamental class of geodesics on any closed surface of genus greater than one. Trans. Am. Math. Soci. 26(1), 25–60 (1924)

    Article  MathSciNet  Google Scholar 

  21. Munkres, J.: Topology. featured titles for topology series (2000)

  22. Devin, M.: Topology and dynamics of the contracting boundary of cocompact CAT (0) spaces. Pac. J. Math. 299(1), 89–116 (2019)

    Article  MathSciNet  Google Scholar 

  23. Abhijit, P., Rahul, P.: Acylindrical Hyperbolicity of Subgroups. arXiv preprint arXiv:1903.00628 (2019)

  24. Yulan, Q., Abdul, Z., Rank One Isometries in Sublinearly Morse Boundaries of CAT (0) Groups. arXiv preprint arXiv:1911.03296 (2019)

  25. Tran, H.C.: Divergence spectra and Morse boundaries of relatively hyperbolic groups. Geometriae Dedicata 194(1), 99–129 (2018)

    Article  MathSciNet  Google Scholar 

  26. Tran, H.: On strongly quasiconvex subgroups. Geom. Topol. 23(3), 1173–1235 (2019)

    Article  MathSciNet  Google Scholar 

  27. Abdalrazzaq, Zalloum.: A symbolic coding of the Morse boundary. arXiv preprint arXiv:1811.10383 (2018)

Download references

Acknowledgements

I would like to thank my advisor Ruth Charney for her enthusiasm and for encouraging me to write this paper.

Author information

Authors and Affiliations

  1. Brandeis University, waltham, USA

    Qing Liu

Authors
  1. Qing Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qing Liu.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Q. Dynamics on the Morse boundary. Geom Dedicata 214, 1–20 (2021). https://doi.org/10.1007/s10711-021-00600-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-021-00600-7

Keywords

Search

Navigation