Geodesic
Monomorphisms of Free Burnside Groups
Matematičeskie zametki, Tome 86 (2009) no. 4, pp. 483-490.

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In the paper, it is proved that, for any odd $n\ge1039$, 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 $n$, $n\ge1039$.
Keywords: absolutely free group, free Burnside group, uniformly nonamenable group, residually finite group, $2$-generated subgroup, Tarski monster, Hopfian group. 
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V. S. Atabekyan. Monomorphisms of Free Burnside Groups. Matematičeskie zametki, Tome 86 (2009) no. 4, pp. 483-490. http://geodesic.mathdoc.fr/item/MZM_2009_86_4_a0/

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