Let $F$ be a free group on $d$ generators. 


Denote by $F\_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F\_k$. What is the normal subgroup growth of $G$?
Explicitly, for each natural number $n$, what is the number of normal subgroups of index not greater than $n$ in $G$? An asymptotic result will also be interesting of course.


For example, the case $k=2$ is covered in the book about subgroup growth.


More generally, Let $C$ be a formation of finite groups ($C$ is closed for quotients and subdirect products). Let $G$ be the quotient of $F$ by the intersection of all normal subgroups $N \unlhd F$ for which $F/N$ is a $C$-group. Assume that $G$ is not trivial ($F$ is not residually $C$). The question is the same as before: What is the normal subgroup growth of $G$?



Normal Subgroup Growth
As is stated in Lubotzky-Segal's book on this topic, very little is known about normal subgroup growth for soluble groups. However, we can say that a rank d free $k$-step solvable group, $k \geq 2$, has normal subgroup growth that is very close to $n^{\log(n)}$. Indeed, this group maps onto any metabelian group generated by <= d generators, and it is known that metabelian groups have growth bounded below by $n^{b(log(n))^{1-2/\kappa(G)}}$ (here $\kappa(G)$ is the Krull dimension, see Theorem 9.2 in Lubotzky-Segal). This gives a lower bound. For the upper bound, use Corollary 2.8 as in my comment. 


I believe we can sharpen the above argument a bit to demonstrate that normal subgroup growth can distinguish $F/F\_3$ from $F/F\_2$. Indeed, if $F$ is a free group of rank greater than 2, then $F/F\_3$ maps onto $(\mathbb{Z} \wr \mathbb{Z}^{(k)}) \rtimes C\_k$ (where the action of $C\_k$ in the semidirect product is the natural one) for any natural number $k$. The group $\mathbb{Z} \wr \mathbb{Z}^{(k)}$ has Krull dimension $k+1$, and so we can, from this, demonstrate that the normal subgroup growth of $F/F\_3$ is never bounded below (asymptotically) by $n^{a (\log(n))^{1-1/t}}$ for any natural number $t$. However, Theorem 9.2 in Lubotzky-Segal shows that this is true for $F/F\_2$.


