Classifying group actions on hyperbolic spaces

[2504.21203] Classifying group actions on hyperbolic spaces Skip to main content Search Submit Donate Log in Mathematics > Group Theory arXiv:2504.21203 (math) [Submitted on 29 Apr 2025 ( v1 ), last revised 28 Aug 2026 (this version, v3)] Title: Classifying group actions on hyperbolic spaces Authors: D. Osin , K. Oyakawa , A. Rapinchuk View PDF HTML (experimental) Abstract: For a given group $G$, it is natural to ask whether one can classify all isometric $G$-actions on Gromov hyperbolic spaces. We propose a formalization of this problem utilizing the complexity theory of Borel equivalence relations. In this paper, we focus on actions of general type, i.e., non-elementary actions without fixed points at infinity. Our main result is the following dichotomy: for every countable group $G$, either all general type actions of $G$ on hyperbolic spaces can be classified by an explicit invariant ranging in an infinite dimensional projective space or they are unclassifiable in a very strong sense. In terms of Borel complexity theory, we show that the equivalence relation associated with the classification problem is either smooth or $K_\sigma$ complete. Special linear groups $SL_2(F)$, where $F$ is a countable field of characteristic $0$, satisfy the former alternative, while non-elementary hyperbolic (and, more generally, acylindrically hyperbolic) groups satisfy the latter. In the course of proving our main theorem, we also obtain results of independent interest that offer new insights into algebraic and geometric properties of groups admitting general type actions on hyperbolic spaces. Comments: Main paper by D. Osin and K. Oyakawa, appendix by D. Osin and A. Rapinchuk. v2: Appendix is added; some results are strengthened. v3: Theorem enumeration in the appendix is fixed. To appear in Adv. Math Subjects: Group Theory (math.GR) ; Geometric Topology (math.GT); Logic (math.LO) Cite as: arXiv:2504.21203 [math.GR] (or arXiv:2504.21203v3 [math.GR] for this version) https://doi.org/10.48550/arXiv.2504.21203 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Denis Osin [ view email ] [v1] Tue, 29 Apr 2025 22:18:52 UTC (130 KB) [v2] Mon, 20 Jul 2026 19:44:39 UTC (92 KB) [v3] Fri, 28 Aug 2026 01:51:58 UTC (92 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GR < prev | next > new | recent | 2025-04 Change to browse by: math math.GT math.LO References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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