Let $F$ be a free group generated by a set $S$. For $g\in F$, let $l(g)$ be the length of $g$ with respect to $S$.
Now for $a\in G$ and $g_1,\dotsc,g_n\in G$, let $T=g_1^{-1}ag_1g_2^{-1}ag_2\dotsm g_n^{-1}ag_n$. What can we say about $l(T)$ in terms of $l(a)$? Assuming that $a$ is cyclically reduced, is it known that $l(T)\geq l(a)$? (I conjectured this, but can't prove it.)
Back ground: The length $l(a^n)$ in terms of $l(a)$ is easily determined. Products of conjugates of $a$ is a natural generalization of powers of $a$. Thus it's natural to ask the relation of the lengths with respect to this generalization.
Presuming that $G=F$: For a counterexample when $n=1$, take $$S = \{s,t\}, \quad a = s^{100} t s^{-100}, \quad g_1 = s^{100} $$ so $$l(a) = 201 \quad\text{and}\quad l(g_1^{-1} a g_1) = 1 $$