Uniform nonamenability of subgroups of free Burnside groups of odd period

Abstract

A famous theorem of Adyan states that, for any m ≥ 2 and any odd n ≥ 665, the freemgenerated 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 ≥ 1003 is a uniformly nonamenable group. This result implies the affirmative answer, for odd n ≥ 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).