Property (NL) for group actions on hyperbolic spaces

TL;DR

This paper introduces Property (NL), characterizing groups that cannot act on hyperbolic spaces with loxodromic elements, and explores its implications for group actions, embeddings, and fixed point properties.

Contribution

It formally defines Property (NL), proves a dynamical criterion for it, and studies its stability and connections to fixed point properties, including applications to Thompson-like groups.

Findings

Many groups satisfy Property (NL).

Groups with rich actions on compact spaces have Property (NL).

Finitely generated groups can embed into simple groups with Property (NL).

Abstract

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.