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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 \(B(m,n)\).

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
Full Text: DOI
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