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Boundaries and equivariant maps for ergodic groupoids

Part of: Other generalizations of groups Ergodic theory Markov processes

Published online by Cambridge University Press:  14 July 2025

Filippo Sarti Open the ORCID record for Filippo Sarti [Opens in a new window]  and
Alessio Savini
Filippo Sarti*
Affiliation:
Department of Mathematics, University of Pisa, Pisa, Italy
Alessio Savini
Affiliation:
Department of Mathematics, University of Milano Bicocca, Milano MI, Italy
*
Corresponding author: Filippo Sarti; Email: filosarti@gmail.com
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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.

Keywords

measured groupoidspoisson boundaryboundary map

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 20N02: Sets with a single binary operation (groupoids) 37A40: Nonsingular (and infinite-measure preserving) transformations

Information

Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 32
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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