  We introduce Property (NL), which indicates that a group does not admit any
(isometric) action on a hyperbolic space with loxodromic elements. In other
words, such a group $G$ can only admit elliptic or horocyclic hyperbolic
actions, and consequently its poset of hyperbolic structures $\mathcal{H}(G)$
is trivial. It turns out that many groups satisfy this property; and we
initiate the formal study of this phenomenon. Of particular importance is the
proof of a dynamical criterion in this paper that ensures that groups with rich
actions on compact Hausdorff spaces have Property (NL). These include many
Thompson-like groups, such as $V, T$ and even twisted Brin--Thompson groups,
which implies that every finitely generated group quasi-isometrically embeds
into a finitely generated simple group with Property (NL). We also study the
stability of the property under group operations and explore connections to
other fixed point properties. In the appendix (by Alessandro Sisto) we describe
a construction of cobounded actions on hyperbolic spaces starting from
non-cobounded ones that preserves various properties of the initial action.
