Twisted conjugacy in PL-homeomorphism groups of the circle

Given an automorphism ϕ : Γ → Γ of a group, one has a left action of Γ on itself defined as g . x = g x ϕ ( g - 1 ) . The orbits of this action are called the Reidemeister classes or ϕ -twisted conjugacy classes. We denote by R ( ϕ ) ∈ N ∪ { ∞ } the Reidemeister number of ϕ , namely, the cardinality of the orbit space R ( ϕ ) if it is finite and R ( ϕ ) = ∞ if R ( ϕ ) is infinite. The group Γ is said to have the R ∞ -property if R ( ϕ ) = ∞ for all automorphisms ϕ ∈ Aut ( Γ ) . We show that the generalized Thompson group T ( r , A , P ) has the R ∞ -property when the slope group P ⊂ R > 0 × is not cyclic.