Hausdorff dimension and the dynamics of diffeomorphisms

A proof of the Hilbert-Smith conjecture for a free Lipschitz action is given. The proof is elementary in the sense that it does not rely on Yang's theorem about the cohomology dimension of the orbit space of the p-adic action. The result turns out to be true for the class of spaces of finite Hausdorff volume, which is considerably wider than Riemannian manifolds. As a corollary to the Lipschitz version of the Hilbert-Smith conjecture, the theorem asserting that the diffeomorphism group of a finite-dimensional manifold has no small subgroups is obtained.