The simplicity of certain groups of subanalytic diffeomorphisms

Let M   be a connected real analytic manifold. We denote by Diffsubr(M)0, 1⩽r<∞1⩽r<∞, the group of subanalytic CrCr diffeomorphisms of M   which are isotopic to the identity via a compactly supported subanalytic CrCr isotopy. We show that Diffsubr(M)0 satisfies Epstein's axioms. This implies that the commutator subgroup of Diffsubr(M)0 is simple. Moreover, we show that the commutator subgroup of Diffsubr(M)0 is dense in Diffsubr(M)0. As a corollary we obtain that Diffsubr(M)0 is topologically simple.