Bowditch representations in Gromov-hyperbolic spaces : characterizations, dynamics of $\mathrm{Out}(\mathbb{F}_2)$ and recognition

We study a generalization of the $BQ$-conditions, introduced by Bowditch and further developed by Tan-Wong-Zhang, for representations of the free group of rank two into isometry groups of Gromov-hyperbolic spaces. We show the existence of an explicit constant $K_\delta$, depending only on the hyperbolicity constant $\delta$ of the space, such that the hyperbolicity of the images of primitive elements together with the finiteness of the set of (conjugacy classes of) primitive elements whose images have lengths bounded by $K_\delta$ imply a linear growth of the lengths with respect to the word length on primitive elements. We give several characterizations of Bowditch representations, and the framework developed allows us to prove that they form an open domain of discontinuity in the character variety. As a corollary, we also obtain a new characterization of primitive-stable representations, introduced by Minsky. Finally, we explain how our results can be used to obtain finite certificate for the recognition of Bowditch representations.