Groups that (do not) act isometrically on hyperbolic spaces

In this paper, we show that if a group acts isometrically on a good hyperbolic space of finite volume entropy through a non-elementary action, then it admits an affine action on some $L^p$ -space with an unbounded orbit for sufficiently large $p$. As an application, we prove that any isometric action of a group with the fixed point property $F_{\mathrm{\infty }}$ on a good hyperbolic space must have a bounded orbit.