A well-known property on groups acting on trees is:
Theorem: Let $T$ be a tree and $g,h \in \mathrm{Isom}(T)$ two elliptic isometries. If $\mathrm{Fix}(g) \cap \mathrm{Fix}(h) = \emptyset$ then the product $gh$ is a loxodromic isometry.
I am pretty sure that a similar statement exists for $\delta$-hyperbolic spaces, where $\mathrm{Fix}$ is replaced with $\mathrm{Fix}_{\delta}$, but I did not succeed in remembering in which article I saw it...
Does somebody know a reference?