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Uniform nonamenability of subgroups of free Burnside groups of odd period. (English. Russian original) Zbl 1213.20036
Math. Notes 85, No. 4, 496-502 (2009); translation from Mat. Zametki 85, No. 4, 516-523 (2009).
Summary: A famous theorem of Adyan states that, for any $$m\geq 2$$ and any odd $$n\geq 665$$, 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 $$n\geq 1003$$ is a uniformly nonamenable group. This result implies the affirmative answer, for odd $$n\geq 1003$$, 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 $$B(m,n)$$.

##### MSC:
 20F50 Periodic groups; locally finite groups 20F05 Generators, relations, and presentations of groups 20E07 Subgroup theorems; subgroup growth 43A07 Means on groups, semigroups, etc.; amenable groups 20F06 Cancellation theory of groups; application of van Kampen diagrams 20F69 Asymptotic properties of groups
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