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.

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.