Notions of Anosov representation of relatively hyperbolic groups

We prove that divergent, extended geometrically finite (in the sense of Weisman arXiv:2205.07183) representations can be interpreted as restricted Anosov (in the sense of Tholozan–Wang arXiv:2307.02934) representations over certain flow spaces. We also show that the representations of this type are stable under small type preserving deformations. As an example, we show that a representation induced from a geometrically finite one through a Galois covering, constructed in Tholozan–Wang arXiv:2307.02934, is divergent and extended geometrically finite with a non-homeomorphic boundary extension.