TITLE: A nilpotent quotient of free groups
QUESTION [5 upvotes]: Let $F$ denote the free group on $n$ generators $g_1,\ldots, g_n$. Consider its quotient $Q$ by the universal relation $[x,[x,y]]$ (a "Serre relation" familiar from Lie theory). This group is nilpotent of class $\leq n$. Denote by $Q_k$ the k-th term of its lower central series. It appears that all commutators of length $\geq 3$ are torsion elements in $Q$, while $Q_2/Q_3$ is the abelian group freely generated by $\{ [g_i, g_j], i<j\}$. Is the precise structure of $Q$ known?

REPLY [5 votes]: As Ian Agol indicated in his comment, groups such as $Q$ have been studied for a while. What you wrote is a specific case of a $n$-Engel group with $n = 2$. The case $n=2$ is, for the most part, completely understood. See this page from group props. Also check out Levi's article Groups in which the commutator operation satisfies certain algebraic conditions (J. Indian Math. Soc. (N.S.) 6, (1942). 87–97). Here it is shown that $Q$ has nilpotence class at most $3$. See also Burnside, W.; On Linear Homogeneous Continuous Groups whose Operations are Permutable. (Proc. London Math. Soc. S1-29 no. 1, 325).
For a start on general $n$, see Havas, George; Vaughan-Lee, M. R. 4-Engel groups are locally nilpotent (Internat. J. Algebra Comput. 15 (2005), no. 4, 649–682).