Let $G$ be a finitely generated group.

For a set of generators $B$ of $G$, $\ell_B(x)$ is the length of the smallest sequence of elements(and inverse of the elements) in $B$, such that the product equals $x$. $\ell_B(X) = \sum_{x\in X} \ell_B(x)$.

We want to find some function $f$, such that any set of generators $B'$ and $b\in B$, we have $\ell_{B'}(b)\leq f(\ell_B(B'))$.

For example, if $G=\mathbb{Z}$, $B=\{1\}$, then we have $$ \ell_{B'}(1)\leq \ell_B(B') $$

Indeed, we can pick two relatively prime elements $u,v\in B'$, and solve for $au+bv = 1$ while minimizing $|a|+|b|$. By Bézout's lemma, this can be bounded by $|u|+|v|\leq \ell_B(B')$.

In particular, I would like to see a result like this for $B=\{a,b\}$ and $G$ is the free group generated by $B$.

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