Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, and $\lvert gh \rvert \leq \lvert g \rvert + \lvert h \rvert$. However, it is possible to add a single letter at the end of $g$, obtaining $g'$ within distance at most $1$ from $g$, such that $\lvert g'h\rvert = \lvert g'\rvert + \lvert h \rvert$.
A result of this kind holds for all non-elementary hyperbolic groups $G$. Namely, there exists a constant $\tau>0$, depending only on $G$ and its generating set, such that for any $g,h\in G$ we may find $g'\in B(g,\tau)$ satisfying $\lvert g'h\rvert \geq \lvert g'\rvert + \lvert h \rvert - 2\tau$, i.e. in the product $g'h$ at most $\tau$ pairs of generators cancel. This is not difficult, but also not completely trivial to show. See Lemma 4.4 on page 9 of my preprint for a short proof.
This lemma seems very natural to me, but I couldn't locate it in the literature. Does anybody know a reference for this?
