Simplicity of Certain Groups of Diffeomorphisms

D. Epstein has shown [1] that for quite general groups of homeomorphisms, the commutator subgroup is simple. In particular, let M be a manifold. (In this note, we assume all manifolds are finite dimensional, Hausdorff, class C, and have a countable basis for their topology.) By a C r + mapping (resp. diffeomorphism) we mean a C mapping (resp. diffeomorphism) whose rth derivative is Lipschitz. Let Diff(M, r) (resp. Diff(M, r+)) denote the group of C (resp. C) diffeomorphisms h of M such that there is an isotropy Ht of h to the identity, and a compact set K such that Ht(x)=x if x e M—K. Epstein showed in [1] that the commutator subgroup of Diff(M, r) (resp. Diff(M, r+)) is simple. Thurston announced in [4] that Diff(Af, oo) is simple. Let n=dim M. In this note we announce the following two results.