Let $F$ be a free group on $\{x_1,x_2,\cdots\}$, $w$ a word in $F$. Then $w$ is a finite expression in $x_i$'s and their inverses. By cancellation, $w$ can be reduced to simplest expression.
Let $w=x_{i_1}x_{i_2}\cdots x_{i_r}$ be a reduced word in $F$. It may happen that end letters can be inverses of each other, i.e. $x_{i_r}=x_{i_1}^{-1}$. In this case, $w$ can be written as
$$w=x_{i_1} (x_{i_2}\cdots x_{i_{r-1}})x_{i_1}^{-1}.$$
Thus, a representative, namely $(x_{i_2}\cdots x_{i_{r-1}})$, of conjugacy class of $w$ is found which has length smaller than that of $w$. Continuing such process, we obtain an expression for $w$ as
$$w=h(x_j x_k\cdots x_l)h^{-1},$$
in which
-
no cancellation can be done (or it is exactly same expression as beginning reduced form)
-
$x_j$ and $x_l$ are not inverses of each other.
This gives a minimum-length representative of conjugacy class of $w$ in $F$.
Question 1. Is this minimum-length conjugacy class representative uniquely determined? (I think, it is unique by the process described).
Question 2. If it is not uniquely determined, then is its length uniquely determined?
I confused with following sentence on Wikipedia (and so posted above questions):
Every word is conjugate to a cyclically reduced word. The cyclically reduced words are minimal-length representatives of the conjugacy classes in the free group. This representative is not uniquely determined, but it is unique up to cyclic shifts (since every cyclic shift is a conjugate element).
Best Answer
Its length if uniquely determined, but for a cyclically reduced word of length $n$ there are at most $n$ minimal length representatives, which you get by cyclically permuting the word.
For example, in the group with free generators $a,b$, the minimal length preresentatives of the conjugacy class of $a^2b^{-1}$ are $a^2b^{-1}$ , $ab^{-1}a$ and $b^{-1}a^2$. These are all conjugate.
However, for $(ab)^2$, there are only two minimal length representatives $(ab)^2$ and $(ba)^2$.
For a general word $w$ of length $n$, if you write $w=v^k$ with $v$ as large as possible, then there are $|v| = n/k$ distinct minimal length representatives.