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 \cite{BF14} for locally compact groups. In the case of a semidirect groupoid $\mathcal{G}=Γ\ltimes X$ obtained by a probability measure preserving action $Γ\curvearrowright X$ of a locally compact group, we show that a boundary pair is exactly $(B_- \times X, B_+ \times X)$, where $(B_-,B_+)$ is a boundary pair for $Γ$. For any measured groupoid $(\mathcal{G},ν)$, 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 \cite{BF:Unpub}, we define algebraic representability for an ergodic groupoid $(\mathcal{G},ν)$. In this way, given any measurable representation $ρ:\mathcal{G} \rightarrow H$ into the $κ$-points of an algebraic $κ$-group $\mathbf{H}$, we obtain $ρ$-equivariant maps $\mathcal{B}_\pm \rightarrow H/L_\pm$, where $L_\pm=\mathbf{L}_\pm(κ)$ for some $κ$-subgroups $\mathbf{L}_\pm<\mathbf{H}$. In the particular case when $κ=\mathbb{R}$ and $ρ$ is Zariski dense, we show that $L_\pm$ must be minimal parabolic subgroups.