The Inner Amenability of the Generalized Thompson Group

In this paper we prove that the general version, F(N) of the Thompson group is inner amenable. As a consequence we generalize a result of P.Jolissaint. To do so, we prove first that F(N) together with a normal subgroup are i.c.c (infinite conjugacy classes) groups. Then, we investigate the relative McDuff property out of which we extract property $\mathrm{\Gamma }$ for the group von Neumann algebras involved. By a result of E.G.Effros, F(N) follows inner amenable.