Takashi Tsuboi

On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds

We show that the identity component Diff(M)0 of the group of Cr diffeomorphisms of a compact (2m)-dimensional manifold M2m (1 ≤ r ≤ ∞, r = 2m + 1) is uniformly perfect for 2m ≥ 6, i.e., any element of Diff(M)0 can be written as a product of a bounded number of commutators. It is also shown that for a compact connected manifold M2m (2m ≥ 6), the identity component Diff(M)0 of the group of Cr diffeomorphisms of M2m (1 ≤ r ≤ ∞, r = 2m + 1) is uniformly simple, i.e., for elements f and g of Diff(M)0 \ {id}, f can be written as a product of a bounded number of conjugates of g or g−1.

Homeomorphism groups of commutator width one

We show that every element of the identity component Homeo(S)0 of the group of homeomorphisms of the n-dimensional sphere Sn can be written as one commutator. We also show that every element of the group Homeo(μn) of homeomorphisms of the n-dimensional Menger compact space μn can be written as one commutator.