gr.group theory - Stable w-length

Here are some weak observations that don't quite answer any of your questions. Let $g$ be a positive integer, and consider the free group $F_{2g}$ generated by $a_k$ and $b_k$ for $k = 1$ to $g$. Consider the word:

$$w_g = [a_1,b_1][a_2,b_2][a_3,b_3] ldots [a_g,b_g].$$

Suppose that $lambda_g = sl(w_g,w_g)$. I claim that for any $x$ in the commutator of $F_2$ with $cl(x) = g$, the stable commutator length $scl(x)$ of $x$ is $le g cdot lambda_g$. Suppose otherwise. First of all, note that for large $n$ we can write $w^n_g$ as the product of (roughly) $n cdot lambda_g cdot g$ commutators. Since the commutator length of $x$ is $g$, there exists a map from $F_{2g}$ to $F_{2}$ such that the image of $w_g$ is $x$. On the other hand, we see that the image of $w^n_g$ is $x^n$, and thus the commutator length of $x^n$ is (asymptotically) at most by $n cdot lambda_g cdot g$, and thus $scl(x) le lambda_g cdot g$.

Example: $cl([x,y]^3) = 2$ and $scl([x,y]^3) = 3/2$, and thus $lambda_3 ge 3/4$. In general, the fact that $scl([x,y]) = 1/2$ implies that that $lambda_g$ tends to one as $g$ increases.

I think one can promote this example to a word in $F_2$. Consider the characteristic homomorphism $phi_n:F_2 rightarrow mathbf{Z} oplus mathbf{Z}
rightarrow mathbf{Z}/nmathbf{Z} oplus mathbf{Z}/nmathbf{Z}$. Suppose that $n$ is odd, and write $2g = n^2 + 1$. The kernel of $F_2$ is free of rank $2g$. Pick generators for $ker(phi_n)$ once and for all, and call them $a_k$ and $b_k$ for $k = 1$ to $g$. We may think of $a_k$ and $b_k$ as elements in $F_2$, but also as formal words. Since $ker(phi_n) = F_{2g}$ is characteristic, the formal words $a_k$ and $b_k$ always yield elements of $F_{2g}$ (alternatively, the images of $a_k$ and $b_k$ in $mathbf{Z} oplus mathbf{Z}$
are divisible by $n$, and this will be so for any substitution of elements of $F_2$ for the generators). Let

$$w_g = [a_1,b_1][a_2,b_2] ldots [a_g,b_g].$$

The argument proceeds as above. If $sl(w_g,w_g) = mu_g$, then we can write $w^n_g$ (for large $n$) as the product of $n cdot mu_g cdot g$ commutators, each of which is the commutator of a pair of elements of $F_{2g}$ (by the characteristic property of the words $a_k$ and $b_k$ described above). Hence, choosing an appropriate map from $F_{2g}$ to $F_2$, we may deduce that for any $x in F_2$ with $cl(x) = g$ that $scl(x) le g cdot mu_g$. Thus we have found words $w_g$ in $F_2$ such that $sl(w_g,w_g)$ tends to $1$ as $g$ goes to infinity. Of course, this says nothing about whether $sl(w_g,w_g)$ actually equals $1$ for any $g$.

Finally, a random other example. If $w = [a,b^2]$, then

$$w^3 = [ab^2a^{-1},b^{-2} ab^2a^{-2}][b^{-2} a b^2,b^4] = [b^{-2} a b^2 a^{-2},(aba^{-1})^2]^{-1}[b^{-2} a b^2,(b^2)^2],$$

so $sl(w,w) le 2/3$.

I wrote this on a very old computer that was too slow for previewing LaTeX, but hopefully this can still be read.