TITLE: A group, neither amenable, nor having a subgroup that looks like $F_2$ up to level $n$?
QUESTION [6 upvotes]: 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. 
But is there a non-amenable group $G$ where for some $n$, $G$ is length-$n$ unfree in the following sense?

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 $G$.

REPLY [12 votes]: 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).