ON THE UNIFORM SIMPLICITY OF DIFFEOMORPHISM GROUPS

We show the uniform simplicity of the identity component Diff(r)(M(n))(0) of the group of C(r) diffeomorphisms Diff(r)(M(n)) (1 <= r <= infinity, r not equal n + 1) of the compact connected n-dimensional manifold M(n) with handle decomposition without handles of the middle index n/2. More precisely, for any elements f and g of such Diff(r)(M(n))(0) \ {id}, f can be written as a product of at most 16n+28 conjugates of g or g(-1), which we denote by f is an element of (C(g))(16n+28). We have better estimates for several manifolds. For the n-dimensional sphere S', for any elements f and g of Diff(r)(S(n))(0) \ {id} (1 <= r <= infinity, r not equal n + 1), f is an element of (C(g))(12), and for a compact connected 3-manifold M(3), for any elements f and g of Diff(r)(M(3))(0) \ {id} (1 <= r <= infinity, r not equal 4): f is an element of (C(g))(44).