Largest quotient of solvability length 2 I've been thinking about quotients of groups, and I know that the abelianization G/[G,G] is important, but what about other quotients? Specifically, what about the quotient G/[G,[G,G]]? Is that important? More specifically I want to know about the quotient G/[G,[G,G]] if G is the free group on two generators. Is there a name for it?
Edit: Silly me, I just worked out this is just the discrete heisenberg group. The group G/[[G,G],[G,G]] is the group I want to know about now.
 A: Given any set of words $\mathfrak{W}$, and a group $G$, the subgroup $\mathfrak{W}(G)$ generated by all values of the words in $\mathfrak{W}$ evaluated at elements of $G$ is a fully invariant subgroup of $G$ called the $\mathfrak{W}$-verbal subgroup of $G$. It is the smallest normal subgroup $N$ of $G$ such that $G/N$ satisfies the words in $\mathfrak{W}$, i.e., such that $G$ lies in the variety of groups determined by $\mathfrak{W}$.
For instance, if $\mathfrak{W}=\{ xyx^{-1}y^{-1}\} = \{[x,y]\}$, then $\mathfrak{W}$ determines the variety of abelian groups, and $\mathfrak{W}(G)=[G,G]$ is the commutator subgroup. If $\mathfrak{W}=\{ [x_1,\ldots,x_{n+1}]\}$, then $\mathfrak{W}$ determines the variety of nilpotent groups of class at most $n$, and $\mathfrak{W}(G) = G_{n+1}$, the $n+1$st term of the lower central series.
If $G$ is the free group on $k$ generators, then $G/\mathfrak{W}(G)$ is the relatively free $\mathfrak{W}$-group of rank $k$. It has the same universal property as the free group of rank $k$, but relative to groups in the variety determined by $\mathfrak{W}$.
Added: If we take the word $[[x,y],[z,w]]$, then this determines the variety of metabelian groups, $\mathfrak{A}^2$ (equivalently: solvable groups of solvability length at most two). The verbal subgroup corresponding to $[[x,y],[z,w]]$ is $G^{(2)}$, the second derived subgroup of $G$. If ${F}$ is the free group of rank $2$, then $F/[[F,F],[F,F]] = F/F^{(2)}$ is the relatively free metabelian group of rank $2$; it has the universal property that given any solvable group $K$ with solvability length at most $2$, and any elements $a,b\in K$, there exists a unique group homomorphism $F/F^{(2)} \to K$ that maps the free generators $x$, $y$ to $a$ and $b$, respectively.
For much more on this, see Hanna Neumann's book Varieties of Groups. 
