Geodesic
Property $\mathrm{(NL)}$ for group actions on hyperbolic spaces (with an appendix by Alessandro Sisto)
Sahana H. Balasubramanya1 orcid ; Francesco Fournier-Facio2 orcid ; Anthony Genevois3 orcid
1State University of New York at Buffalo, USA; Lafayette College, Easton, USA
2ETH Zürich, Zurich, Switzerland; University of Cambridge, UK
3Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, France
Groups, geometry, and dynamics, Tome 19 (2025) no. 4, pp. 1319-1371

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DOI

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 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.
DOI : 10.4171/ggd/806
Classification : 20F65, 20F67, 37B05
Mots-clés : geometric group theory, hyperbolic actions, loxodromic elements, quasimorphisms, Thompson-like groups

Sahana H. Balasubramanya  1   ; Francesco Fournier-Facio  2   ; Anthony Genevois  3

1 State University of New York at Buffalo, USA; Lafayette College, Easton, USA
2 ETH Zürich, Zurich, Switzerland; University of Cambridge, UK
3 Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, France 
Sahana H. Balasubramanya; Francesco Fournier-Facio; Anthony Genevois. Property $\mathrm{(NL)}$ for group actions on hyperbolic spaces (with an appendix by Alessandro Sisto). Groups, geometry, and dynamics, Tome 19 (2025) no. 4, pp. 1319-1371. doi: 10.4171/ggd/806
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     title = {Property $\mathrm{(NL)}$ for group actions on hyperbolic spaces (with an appendix by {Alessandro} {Sisto)}},
     journal = {Groups, geometry, and dynamics},
     pages = {1319--1371},
     year = {2025},
     volume = {19},
     number = {4},
     doi = {10.4171/ggd/806},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/806/}
}
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