Property for group actions on hyperbolic spaces (with an appendix by Alessandro Sisto)
Sahana H. Balasubramanya
State University of New York at Buffalo, Buffalo, USA; Lafayette College, Easton, USAFrancesco Fournier-Facio
ETH Zürich, Zurich, Switzerland; University of Cambridge, Cambridge, UKAnthony Genevois
Université de Montpellier, Montpellier, France
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Abstract
We introduce Property , which indicates that a group does not admit any (isometric) action on a hyperbolic space with loxodromic elements. In other words, such a group can only admit elliptic or horocyclic hyperbolic actions, and consequently its poset of hyperbolic structures 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 . These include many Thompson-like groups, such as and even twisted Brin–Thompson groups, which implies that every finitely generated group quasi-isometrically embeds into a finitely generated simple group with Property . 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.
Cite this article
Sahana H. Balasubramanya, Francesco Fournier-Facio, Anthony Genevois, Property for group actions on hyperbolic spaces (with an appendix by Alessandro Sisto). Groups Geom. Dyn. (2024), published online first
DOI 10.4171/GGD/806