Let $F\_m$ be the free group with $m$ generators $S:=\{x\_1,\dots, x\_m\}$. I am interested in the following quantity 
$$
F(n):=\frac{|\{w\in [F\_m,F\_m]: \|w\|\_S\leq n\}|}{|B\_S(n)|}
$$
where $B\_S(n):=\{w\in F\_m: \|w\|\_S\leq n\}$ and $\|w\|\_S$ is the word metric of $w$, and $[F\_m,F\_m]$ is the commutator subgroup of $F\_m$. Do we know how $F(n)$ growth as a function of $n$? Presumably this a classic problem in group theory. 


I would be thankful if you please mention a reference that consider this function. 



Commutators in free groups
The question is indeed very close to the one about the return probabilities on the abelian group $\mathbb Z^m$. Still, as Benjamin has noticed, there is some difference, since here one has to consider the non-backtracking paths only. 


There is a standard way to deal with it which is based on the so-called **random walks with internal degrees of freedom** (also known under numerous other names). In this concrete context this is the Markov chain on the product of the free group $F\_m$ by the set $A$ of generators and their inverses with the transition probabilities 
$$
p( (g,a), (ga',a') ) = 1/(2m-1) \;, \qquad a'\neq a^{-1} \;.
$$
This particular chain is called **non-backtracking simple random walk** in the paper MR2342439 (2007) by Kaimanovich, Kapovich and Schupp. Its sample paths are precisely non-backtracking paths in the free group, and its time $n-1$ distribution for the initial distribution uniform on the set of $(a,a):a\in A$ is precisely the uniform measure on the words of length $n$ in the free group.


In the same way one can define the quotient of the above chain on $\mathbb Z^m$. Then the original question is precisely the question about an asymptotic of transition probabilities of this quotient chain. In the abelian case such chains have been extensively studied since early 80s, one of the earliest papers being the one by Krámli and Szász MR0699788 (1983). The asymptotic the OP is asking about is of course $n^{-m/2}$, which was later rediscovered and generalized in numerous other publications. However, I will refrain from going into the detailed history.


