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 $\mathcal{G}=\hat{I}“⋉X$ obtained by a probability measure preserving action $\hat{I}“\curvearrowright X$ 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 $\hat{I}“$. For any measured groupoid $(\mathcal{G},\hat{I}½)$, we prove that the Poisson boundaries associated to the Markov operators generated by a probability measure equivalent to $\hat{I}½$ provide other examples of our definition. Following Bader and Furman \cite{BF:Unpub}, we define algebraic representability for an ergodic groupoid $(\mathcal{G},\hat{I}½)$. In this way, given any measurable representation $\ddot{I}:\mathcal{G}\to H$ into the $\hat{I}º$-points of an algebraic $\hat{I}º$-group $\mathbf{H}$, we obtain $\ddot{I}$-equivariant maps ${\mathcal{B}}_{\pm }\to H/L_{\pm }$, where $L_{\pm }={\mathbf{L}}_{\pm }(\hat{I}º)$ for some $\hat{I}º$-subgroups ${\mathbf{L}}_{\pm }<\mathbf{H}$. In the particular case when $\hat{I}º=\mathbb{R}$ and $\ddot{I}$ is Zariski dense, we show that $L_{\pm }$ must be minimal parabolic subgroups.

39 pages, final version to appear in Glasgow Mathematical Journal