Twisted conjugacy in PL-homeomorphism groups of the circle

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(r, A, P) has the $R_\infty$-property when the slope group $P\subset {\mathbb {R}}^\times _{>0}$ is not cyclic.