Uniform perfectness of diffeomorphism groups, I

Let D(M) be the group of diffeomorphisms of a smooth manifold M which  are isotopic to the identity through diffeomorphisms with compact  support. Herman and Thurston proved that D(M) is perfect which means  that D(M) coincides with its commutator subgroup. D(M) is said to be  uniformly perfect if each element of D(M) can be written as a product  of a bounded number of commutators. Hermann, Burago-Ivanov-Polterovich  and Tsuboi have proved that D(M) is uniformly perfect if dim(M) is not  equal to 2 or 4. There are many results on uniform perfectness for  various groups.

Let N be a smooth submanifold of M and let D(M;N) denote the group of  diffeomorphisms of M preserving N which are isotopic to the identity  through diffeomorphisms preserving N of with compact support. We can  prove that the group D(M;N) is perfect if dim(N) is positive.

In this talk we consider the uniform perfectness of D(M;N) for the  case of dim(N) positive. D(M;N) is not always uniformly perfect. We  shall find the conditions for D(M;N) to be uniformly perfect. We  explain that it depends
on the geometric structure of M and N.