2024/02/23 by Filippo Sarti, Sarti, Filippo, Alessio Savini +1
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2402.15355
openalex publication_date 2024/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a notion of boundary pair (B-,B+) for measured groupoids which generalizes the one introduced by Bader and Furman \citeBF14 for locally compact groups. In the case of a semidirect groupoid G=Γ\ltimes X obtained by a probability measure preserving action Γ\curvearrowright X of a locally compact group, we show that a boundary pair is exactly (B- × X, B+ × X), where (B-,B+) is a boundary pair for Γ. For any measured groupoid (G,ν), we prove that the Poisson boundaries associated to the Markov operators generated by a probability measure equivalent to ν provide other examples of our definition. Following Bader and Furman \citeBF:Unpub, we define algebraic representability for an ergodic groupoid (G,ν). In this way, given any measurable representation ρ:G → H into the κ-points of an algebraic κ-group H, we obtain ρ-equivariant maps B_± → H/L_±, where L_±=L_±(κ) for some κ-subgroups L_±<H. In the particular case when κ=ℝ and ρ is Zariski dense, we show that L_± must be minimal parabolic subgroups.