Hyperfiniteness of the boundary action of virtually special groups

2025/09/04 by Koichi Oyakawa, Oyakawa, Koichi
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2509.04613

openalex publication_date 2025/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite. This can be considered as a generalization of hyperfiniteness of the boundary action of cubulated hyperbolic groups by Huang-Sabok-Shinko.

Citations

• Graphical small cancellation and hyperfiniteness of boundary actions
• Borel asymptotic dimension of the Roller boundary of finite dimensional CAT(0) cube complexes
• Hyperfiniteness of boundary actions of acylindrically hyperbolic groups
• Hyperfiniteness and Borel asymptotic dimension of boundary actions of hyperbolic groups
• Hyperfiniteness of boundary actions of relatively hyperbolic groups
• Realizations of countable Borel equivalence relations
• Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type
• The geometry of the curve graph of a right-angled Artin group
• The virtual Haken conjecture
• The Poisson boundary of CAT(0) cube complex groups
• Complexes and Exactness of certain Artin Groups
• Amenability of actions on the boundary of a building
• Geometry of the mapping class groups I: Boundary amenability
• Boundary Amenability of Relatively Hyperbolic Groups
• Boundary amenability of hyperbolic spaces

Related

• Hyperfiniteness of boundary actions of relatively hyperbolic groups
• Hyperfiniteness and Borel asymptotic dimension of boundary actions of hyperbolic groups
• A boundary criterion for cubulation
• Hyperbolicities in CAT(0) cube complexes
• Comparing the Roller and B(X) boundaries of CAT(0) cube complexes