THE VON NEUMANN ALGEBRA OF THE CANONICAL EQUIVALENCE RELATION OF THE GENERALIZED THOMPSON GROUP

We study the equivalence relation RN generated by the (non free) action of the generalized Thompson group F(N) on the unit interval. We show that this relation is a standard, quasipreserving ergodic equivalence relation. Using results of Feldman-Moore, Krieger and Connes we prove that the von Neumann algebra M(R-N) associated to R-N is the hyperfinite type III lambda factor, with lambda = 1/N. Moreover we analyze R-N and F(N) in connection with Gaboriau's work on costs of groups. We prove that the cost C(F(N)) = 1 for any N >= 2 and for N = 2 we precisely find a treeing of R-N.