2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70725We 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.9 pagesGroup Theory20F65; 20F69text