TITLE: A result of Schützenberger on commutators and powers in free groups
QUESTION [17 upvotes]: It is an old result of Schützenberger that in a free group, a basic commutator cannot be a proper power. A look at the original reference
M.-P. Schützenberger, Sur l'équation $a^{2+n} = b^{2+m}c^{2+p}$ dans un groupe libre, C. R. Acad. Sci. Paris 248 (1959), 2435–2436 (French).
quickly reveals that a lot of details are missing and some claims appear to be wrong.

Question: Let $F$ be a free group and $a,b,c \in F$ with $c \neq 1$ and $n \geq 2$. Why is $[a,b] \neq c^n$?

In particular, it would be nice to have a somewhat geometric proof of this apparently fundamental fact. There is an algebraic proof in 
G. Baumslag, Some aspects of groups with unique roots, Acta Math 104(3) (1960), 217–303.
as Lemma 36.4 but it is relies on various technical computations and is hard to grasp.

REPLY [4 votes]: There is a visualisation of Schützenberger's
observation based on a variant of the
car-crash lemma that says that

for any multiple motion
on a map on a closed oriented
surface $S$, the
number of points of complete collision is at least
$\chi(S)+\sum\limits_D(d_D-1)$,
where the summation runs over all
faces $D$ of the map
and $d_D$ is the number of cars moving around a face $D$.

See
[this self-advertisement]
for exact definitions.
So, if a non-identity commutator in a free group $F(x,y,\dots)$ is, e.g.,
a cube of a word $c$, we obtain a one-face map on a torus, where the
label of the face is $c^3$. Let three cars move around this face
counter-clockwise
with a
constant speed one edge per a minute; at a moment
$t$ each car moves along an edge labelled by the $[t]$th letter
of the word $c$ (where the integer part $[t]$
of $t\in\mathbb R$ is counted modulo the length of $c$).

The car-crash lemma asserts that
$$
\small
\pmatrix{
\hbox{the number of points}
\\\\ 
\hbox{of complete collision}
}
\geqslant
\begin{pmatrix}
\hbox{the Euler characteristic}
\\\\ 
\hbox{of the torus,}
\\\\
\hbox{i.e. 0}
\end{pmatrix}
+
\pmatrix{
\hbox{$d_D$, i.e. the number of cars}
\\\\
\hbox{moving around the unique face $D$,}
\\\\
\hbox{i.e. 3}
}
-1=2.
%}
$$
On the other hand, a collision inside an edge is impossible:

such a collision would imply that some letter of the word
$c$ is $x$ and $x^{-1}$ simultaneously. A similar argument shows
that a collision at a vertex cannot occur too. This contradiction
proves that a commutator cannot be a cube.
This approach was used in
[this self-advertisement],
[this self-advertisement],
and
[this self-advertisement]
to obtain generalisations of
Schützenberger's observation
and
Duncan--Howie's
theorem (mentioned by Henry Wilton).