Exercise 3, § 2

Let G be a group equal to its derived group (A, I, p. 67). For every $g\in G$, we note $\ell (g)$ the smallest integer $k\geqslant 0$ such that there exist pairs $(a_1, b_1)$, . . . , $(a_k, b_k)$ in $G\times G$ verifying $g= [a_1, b_1]\cdots [a_k, b_k]$.

a) The mapping $\ell$ is a distance on G.

b) For every $g\in G$, the limit

lsc($g$) $=$ lim$_{n\rightarrow+\infty}\frac{\ell(g^n)}{n}$

exists. It is called the stable length of the commutators of $g$. c) Let S be a set and let G = D(F(S)) be the derived group of the free group generated by S. The group G is equal to its derived group, and for every $g\in G$, $g\not=e$, one has lsc($g$)$\geqslant 1/2$. There exists $g\in G$ such that lsc($g$) $= 1/2$.