Advertisement

Twisted conjugacy in PL-homeomorphism groups of the circle

  • Daciberg Lima Gonçalves
  • Parameswaran Sankaran
Original Paper
  • 10 Downloads

Abstract

Given an automorphism \(\phi :\Gamma \rightarrow \Gamma \) of a group, one has a left action of \(\Gamma \) on itself defined as \(g.x=gx\phi (g^{-1})\). The orbits of this action are called the Reidemeister classes or \(\phi \)-twisted conjugacy classes. We denote by \(R(\phi )\in {\mathbb {N}}\cup \{\infty \}\) the Reidemeister number of \(\phi \), namely, the cardinality of the orbit space \({\mathcal {R}}(\phi )\) if it is finite and \(R(\phi )=\infty \) if \({\mathcal {R}}(\phi )\) is infinite. The group \(\Gamma \) is said to have the \(R_\infty \)-property if \(R(\phi )=\infty \) for all automorphisms \(\phi \in {\text {Aut}}(\Gamma )\). We show that the generalized Thompson group T(rAP) has the \(R_\infty \)-property when the slope group \(P\subset {\mathbb {R}}^\times _{>0}\) is not cyclic.

Keywords

R. Thompson’s groups PL-homeomorphisms of the circle Twisted conjugacy Reidemeister number \(R_\infty \)-property 

Mathematics Subject Classification (2010)

20E45 20E36 

Notes

Acknowledgements

This work was concluded during the visit of the first author to the Institute of Mathematical Sciences, Chennai. He would like to thank the Institute for the great hospitality and work environment. He was partially supported by Projeto Temático Topologia Algébrica, Geométrica e Diferencial FAPESP No. 2016/24707-4. The second author was partially supported by a XII Plan Project, Department of Atomic Energy, Government of India.

References

  1. 1.
    Bieri, R., Strebel, R.: On Groups of PL-Homeomorphisms of the Real Line. Mathematical Surveys and Monographs, vol. 215. American Mathematical Society, Providence (2016)CrossRefGoogle Scholar
  2. 2.
    Bleak, C., Fel’shtyn, A., Gonçalves, D.L.: Twisted conjugacy in R. Thompson’s group \(F\). Pac. J. Math. 238(1), 1–6 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brin, M.G., Guzmán, F.: Automorphisms of generalized Thompson groups. J. Algebra 203, 285–348 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brown, K.S.: Finiteness properties of groups. J. Pure Appl. Algebra 44, 45–75 (1987)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Burillo, J., Matucci, F., Ventura, E.: The conjugacy problem in extensions of Thompson’s group \(F\). Isr. J. Math. 216(1), 15–59 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gonçalves, D.L., Kochloukova, D.H.: Sigma theory and twisted conjugacy classes. Pac. J. Math. 247(2), 335–352 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gonçalves, D.L., Sankaran, P.: Twisted Conjugacy in Richard Thompson’s Group \(T\). arxiv:1309.2875v2 [math.GR]
  8. 8.
    Gonçalves, D.L., Sankaran, P.: Sigma theory and twisted conjugacy classes, II: Houghton groups and pure symmetric automorphism groups. Pac. J. Math. 280(2), 349–369 (2016)CrossRefGoogle Scholar
  9. 9.
    Gonçalves, D.L., Sankaran, P., Strebel, R.: Groups of PL homeomorphisms admitting non-trivial invariant characters. Pac. J. Math. 287(1), 101–158 (2017)CrossRefGoogle Scholar
  10. 10.
    Liousse, I.: Rotation numbers in Thompson–Stein groups and applications. Geom. Dedicata 131, 49–71 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    McCleary, S.: H Groups of homeomorphisms with manageable automorphism groups. Commun. Algebra 6, 497–528 (1978)MathSciNetCrossRefGoogle Scholar
  12. 12.
    McCleary, S., Rubin, M.: Locally moving groups and the reconstruction problems for chains and circles. Preprint, Bowling Green University, Bowling Green, Ohio, 1996. arXiv:math/0510122v1, 6 Oct (2005)
  13. 13.
    Stein, E.M., Shakarchi, R.: Real Analysis. Princeton Lectures in Analysis, vol. 3. Princeton University, Princeton (2005)zbMATHGoogle Scholar
  14. 14.
    Stein, M.: Groups of piecewise linear homeomorphisms. Trans. Am. Math. Soc. 332(2), 477–514 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Daciberg Lima Gonçalves
    • 1
  • Parameswaran Sankaran
    • 2
  1. 1.Department of Mathematics - IMEUniversidade de São PauloSão PauloBrazil
  2. 2.The Institute of Mathematical Sciences, (HBNI)ChennaiIndia

Personalised recommendations