Among the techniques provided by geometric group theory in the study of groups, constructing and studying isometric actions on hyperbolic spaces and their boundaries is one of the most fruitful and has received a lot of attention in the last decades. In this talk, I focus on groups that are not reachable by such a strategy. I will introduce Property (NL), which indicates that a group does not admit any (isometric) action on a hyperbolic space with loxodromic elements. It turns out that many groups satisfy this property; including many Thompson-like groups. In particular, every finitely generated group quasi-isometrically embeds into a finitely generated simple group with Property (NL). I will also talk about the stability of the property under group operations and explore connections to other fixed point properties and the poset of hyperbolic structures. 
This talk is based on a paper co-authored with F.Fournier and A.Genevois, with an appendix by A.Sisto.