Uniform Nonamenability of Subgroups of Free Burnside Groups of Odd Period

A famous theorem of Adyan states that, for any $m\ge 2$ and any odd $n\ge 665$, the free $m$-generated Burnside group $B(m,n)$ of period $n$ is not amenable. It is proved in the present paper that every noncyclic subgroup of the free Burnside group $B(m,n)$ of odd period $n\ge 1003$ is a uniformly nonamenable group. This result implies the affirmative answer, for odd $n\ge 1003$, to the following de la Harpe question: Is it true that the infinite free Burnside group $B(m,n)$ has uniform exponential growth? It is also proved that every $S$-ball of radius $(400n)^3$ contains two elements which form a basis of a free periodic subgroup of rank 2 in $B(m,n)$, where $S$ is an arbitrary set of elements generating a noncyclic subgroup of $B(m,n)$.