Twisted conjugacy in PL-homeomorphism groups of the circle

Given an automorphism phi : Gamma -> 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) is an element of N boolean OR {infinity} the Reidemeister number of phi, namely, the cardinality of the orbit space R(phi) if it is finite and R(phi) = infinity if R(phi) is infinite. The group Gamma is said to have the R-infinity-property if R(phi) = infinity for all automorphisms phi is an element of Aut(Gamma). We show that the generalized Thompson group T (r, A, P) has the R-infinity-property when the slope group P subset of R->0(x) is not cyclic.