Boundaries and equivariant maps for ergodic groupoids

Abstract We give a notion of boundary pair $(\mathcal{B}_-,\mathcal{B}_+)$ for measured groupoids which generalizes the one introduced by Bader and Furman [BF14] for locally compact groups. In the case of a semidirect groupoid $\mathcal{G}=\Gamma \ltimes X$ obtained by a probability measure preserving action $\Gamma \curvearrowright X$ of a locally compact group, we show that a boundary pair is exactly $(B_- \times X, B_+ \times X)$ , where $(B_-,B_+)$ is a boundary pair for $\Gamma$ . For any measured groupoid $(\mathcal{G},\nu )$ , we prove that the Poisson boundaries associated to the Markov operators generated by a probability measure equivalent to $\nu$ provide other examples of our definition. Following Bader and Furman [BF], we define algebraic representability for an ergodic groupoid $(\mathcal{G},\nu )$ . In this way, given any measurable representation $\rho \,:\,\mathcal{G} \rightarrow H$ into the $\kappa$ -points of an algebraic $\kappa$ -group $\mathbf{H}$ , we obtain $\rho$ -equivariant maps $\mathcal{B}_\pm \rightarrow H/L_\pm$ , where $L_\pm =\mathbf{L}_\pm (\kappa )$ for some $\kappa$ -subgroups $\mathbf{L}_\pm \lt \mathbf{H}$ . In the particular case when $\kappa =\mathbb{R}$ and $\rho$ is Zariski dense, we show that $L_\pm$ must be minimal parabolic subgroups.