Finitely generated infinite simple groups of homeomorphisms of the real line

  • James Hyde
  • Yash LodhaEmail author


We construct examples of finitely generated infinite simple groups of homeomorphisms of the real line. Equivalently, these are examples of finitely generated simple left (or right) orderable groups. This answers a well known open question of Rhemtulla from 1980 concerning the existence of such groups. In fact, our construction provides a family of continuum many isomorphism types of groups with these properties.

Mathematics Subject Classification

Primary: 43A07 Secondary: 20F05 



  1. 1.
    Bekka, B., de la Harpe, P., Valette, A.: Kazhdan’s Property (T). Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Belk, J.: Thompson’s Group \(F\). Ph.D. thesis, Cornell University. arXiv:0708.3609 (2004)
  3. 3.
    Bergman, G.: Right orderable groups that are not locally indicable. Pac. J. Math 147(2), 243–248 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonatti, C., Lodha, Y., Triestino, M.: Hyperbolicity as an Obstruction to Smoothability for One-Dimensional Actions. arXiv:1706.05704v3
  5. 5.
    Burillo, J., Lodha, Y., Reeves, L.: Commutators in groups of piecewise projective homeomorphisms. Adv. Math. 332, 34–56 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cannon, J.W., Floyd, W.J., Parry, W.R.: Introductory notes on Richard Thompson’s groups. Enseign. Math. (2) 42(3–4), 215–256 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Clay, A., Rolfsen, D.: Ordered Groups and Topology. American Mathematical Society, Providence (2016). ISBN 1470431068, 9781470431068Google Scholar
  8. 8.
    De La Harpe, P.: Topics in Geometric Group Theory (Chicago Lectures in Mathematics 2000). ISBN 10: 0226317196, ISBN 13: 9780226317199 (2000)Google Scholar
  9. 9.
    Deroin, B., Navas, A., Rivas, C.: Groups, Orders, and Dynamics. arXiv:1408.5805v2
  10. 10.
    Higman, G.: Finitely Presented Infinite Simple Groups. Department of Pure Mathematics, Department of Mathematics, I.A.S., Australian National University, Canberra (1974)Google Scholar
  11. 11.
    Kim, S., Koberda, T., Lodha, Y.: Chain groups of homeomorphisms of the interval. arXiv:1610.04099. (to appear in Ann. Sci. de l’ENS)
  12. 12.
    Lodha, Y.: A Finitely Presented Infinite Simple Group of Homeomorphisms of the Circle. arXiv:1710.06220
  13. 13.
    Lodha, Y., Matte Bon, N., Triestino, M.: Property FW, Differentiable Structures, and Smoothability of Singular Actions. arXiv:1803.08567
  14. 14.
    Martinez, J.: Ordered algebraic structures. In: Proceedings of the Caribbean Mathematics Foundation Conference on Ordered Algebraic Structures, Curacao (1988)Google Scholar
  15. 15.
    Mazurov, V.D., Khukhro, E.I. (eds.): Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version). arXiv:1401.0300v10
  16. 16.
    Morris, D.W.: Amenable groups that act on the line. Algebraic Geom. Topol. 6(2006), 2509–2518 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Navas, A.: Groups, orders, and laws. Groups Geom. Dyn. 8(3), 863–882 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Navas, A.: Group actions on 1-manifolds: a list of very concrete open questions. In: Proceedings of the ICM (2018)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of St. AndrewsSt. AndrewsUK
  2. 2.Institute of MathematicsEPFLLausanneSwitzerland

Personalised recommendations