Uniform Non–amenability of Free Burnside Groups

The aim of the present note is to 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. This answers a question of de la Harpe [12].