It is known that there are non-amenable groups not containing $F\_2$, the free group on two generators. We can even have that [every 2-generated subgroup is finite](https://mathoverflow.net/questions/278391/does-every-non-amenable-group-contain-a-2-generated-non-amenable-subgroup). 


But is there a non-amenable group $G$ where for some $n$, $G$ is length-$n$ unfree in the following sense?



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> **Definition**. $G$ is **length-$n$ unfree** if for all $a,b\in G$, there exist words $u\ne v$ of length $n$ over the alphabet $\{a,b\}$ such that $u=v$ in $G$.
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A group, neither amenable, nor having a subgroup that looks like $F_2$ up to level $n$?
Burnside groups of exponent $n$ are length $n$ unfree. If $n\ge 665$, odd, then the free Burnside group of exponent $n$ of rank 2 or more is not amenable (see Adian's book "The Burnside problem" or Olshanskii's book "Geometry of defining relations" or my book "Combinatorial algebra: syntax and semantics", Chapter 5). 


