Work

Classifying group actions on hyperbolic spaces

Year: 2026

Type: article

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 𝜎 complete. Special linear groups 𝑆 ⁢ 𝐿 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.

Source: Advances in Mathematics

DOI: 10.1016/j.aim.2026.111234

Authors: D. Osin, K. Oyakawa, A. Rapinchuk

Institutions: Vanderbilt University, University of California, Berkeley, University of Virginia

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,790

Funders: National Science Foundation, National Sleep Foundation