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 [BF14] for locally compact groups. In the case of a semidirect groupoid G = X obtained by a probability measure preserving action 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 . 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 [BF], 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.