Boundaries and equivariant maps for ergodic groupoids
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917309944168448 |
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| author | Sarti, Filippo Savini, Alessio |
| author_facet | Sarti, Filippo Savini, Alessio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15355 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundaries and equivariant maps for ergodic groupoids Sarti, Filippo Savini, Alessio Dynamical Systems Probability 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. |
| title | Boundaries and equivariant maps for ergodic groupoids |
| topic | Dynamical Systems Probability |
| url | https://arxiv.org/abs/2402.15355 |