Mathematische Zeitschrift

, Volume 280, Issue 3–4, pp 905–918 | Cite as

Left-orderable groups that don’t act on the line

  • Kathryn MannEmail author


We show that the group \(\mathcal {G}_\infty \) of germs at infinity of orientation-preserving homeomorphisms of \(\mathbb {R}\) admits no action on the line. This gives an example of a left-orderable group of the same cardinality as \({{\mathrm{Homeo}}}_+(\mathbb {R})\) that does not embed in \({{\mathrm{Homeo}}}_+(\mathbb {R})\). As an application of our techniques, we construct a finitely generated group \(\varGamma \subset \mathcal {G}_\infty \) that does not extend to \({{\mathrm{Homeo}}}_+(\mathbb {R})\) and, separately, extend a theorem of E. Militon on homomorphisms between groups of homeomorphisms.


Orderable group Germs of homeomorphisms Homeomorphisms of the line Group actions 

Mathematics Subject Classification

20F60 37E05 58D05 



I thank Andrés Navas for introducing me to this problem and much of the background material. I would also like to thank the University of Santiago for its hospitality during my visit in April 2014, and Sebastian Hurtado, Andrés Navas, and Cristóbal Rivas for many productive conversations during the visit (which inspired this work) and their contributions to this paper. Finally, I thank Danny Calegari, Benson Farb, John Franks, Emmanuel Militon, and Dave Witte-Morris for their feedback and interest in this project, and the referee for numerous small improvements.


  1. 1.
    Deroin, B., Navas, A., Rivas, C.: Groups, Orders, and Dynamics. Preprint. arXiv:1408.5805 (2014)
  2. 2.
    Fine, N., Schweigert, G.: On the Group of Homeomorphisms of an Arc. Ann. Math. 62(2), 237–253 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ghys, E.: Groups acting on the circle. L’Enseign. Math. 47, 329–407 (2001)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Holland, W.C.: The lattice-ordered group of automorphisms of an ordered set. Mich. Math. J. 10, 399–408 (1963)CrossRefzbMATHGoogle Scholar
  5. 5.
    Kopytov, V., Medvedev, N.: Right ordered groups. Siberian School of Algebra and Logic, Plenum Publ. Corp, New York (1996)Google Scholar
  6. 6.
    Matsumoto, S.: Numerical invariants for semiconjugacy of homeomorphisms of the circle. Proc. AMS 98(1), 163–168 (1986)CrossRefzbMATHGoogle Scholar
  7. 7.
    Militon, E.: Actions of groups of homeomorphisms on one-manifolds. Preprint. arXiv:1302.3737 (2013)
  8. 8.
    Navas, A.: On the dynamics of (left) orderable groups. Preprint. arXiv:0710.2466 (2010)

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUC BerkeleyBerkeleyUSA

Personalised recommendations