Boundaries and equivariant maps for ergodic groupoids

We give a notion of boundary pair (B₋,B₊) for measured groupoids which generalizes the one introduced by Bader and Furman for locally compact groups. In the case of a semidirect groupoid G=Γ X obtained by a probability measure preserving action Γ 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, 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.