A well-known property on groups acting on trees is:



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> **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.
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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?



Products of elliptic isometries
Check Lemme 2.3 in chapter 9 of "Geometrie et theorie des groupes" by Coornaert, Delzant and Papadopoulos. It implies that your conclusion holds provided that the quasi-fixed point sets are sufficiently far apart.


