Uniform non--amenability of free Burnside groups

We show that free Burnside groups of sufficiently large odd exponent are non–amenable in a certain strong sense, more precisely, their left regular representations are isolated from the trivial representation uniformly on finite generating sets. This result is applied to the solution of a strong version of the von Neumann – Day problem concerning amenability of groups without non–abelian free subgroups. As another consequence, we obtain that the above–mentioned groups are of uniform exponential growth.