Twisted conjugacy in PL-homeomorphism groups of the circle

  • Daciberg Lima Gonçalves
  • Parameswaran SankaranEmail author
Original Paper


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.


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

Mathematics Subject Classification (2010)

20E45 20E36 



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.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

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

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