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.