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Monomorphisms of free Burnside groups. (English. Russian original) Zbl 1200.20028
Math. Notes 86, No. 4, 457-462 (2009); translation from Mat. Zametki 86, No. 4, 483-490 (2009).
Summary: It is proved that, for any odd $$n\geq 1039$$, there are words $$u(x,y)$$ and $$v(x,y)$$ over the group alphabet $$\{x,y\}$$ such that, if $$a$$ and $$b$$ are any two noncommuting elements of the free Burnside group $$B(m,n)$$, then, for some $$k$$, the elements $$u(a^k,b)$$ and $$v(a^k,b)$$ freely generate a free Burnside subgroup of the group $$B(m,n)$$. In particular, the facts proved in the paper imply the uniform nonamenability of the group $$B(m,n)$$ for odd $$n$$, $$n\geq 1039$$.

##### MSC:
 20F50 Periodic groups; locally finite groups 20F05 Generators, relations, and presentations of groups 20E07 Subgroup theorems; subgroup growth 20E36 Automorphisms of infinite groups 43A07 Means on groups, semigroups, etc.; amenable groups
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