Amenable actions of real and $p$-adic algebraic groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911872694878208 |
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| author | Valette, Alain J. |
| author_facet | Valette, Alain J. |
| contents | Let $K$ be a locally compact field of characteristic 0. Let $G$ be a linear algebraic group defined over $K$, acting algebraically on an algebraic variety $V$. We prove that the action of $G(K)$ (the group of $K$-rational points of $G$) on $V(K)$ is topologically amenable, if and only if all points stabilizers in $G(K)$ are solvable-by-compact. This follows by combining a result by Borel-Serre \cite{BoSe} with the following fact: let $G$ be a second countable locally compact group acting continuously on a second countable locally compact space $Y$. If the action $G\curvearrowright Y$ is smooth (i.e. the Borel structure on $G\backslash Y$ is countably separated), then topological amenability of $G\curvearrowright Y$ is equivalent to amenability of all point stabilizers in $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06094 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Amenable actions of real and $p$-adic algebraic groups Valette, Alain J. Dynamical Systems Let $K$ be a locally compact field of characteristic 0. Let $G$ be a linear algebraic group defined over $K$, acting algebraically on an algebraic variety $V$. We prove that the action of $G(K)$ (the group of $K$-rational points of $G$) on $V(K)$ is topologically amenable, if and only if all points stabilizers in $G(K)$ are solvable-by-compact. This follows by combining a result by Borel-Serre \cite{BoSe} with the following fact: let $G$ be a second countable locally compact group acting continuously on a second countable locally compact space $Y$. If the action $G\curvearrowright Y$ is smooth (i.e. the Borel structure on $G\backslash Y$ is countably separated), then topological amenability of $G\curvearrowright Y$ is equivalent to amenability of all point stabilizers in $G$. |
| title | Amenable actions of real and $p$-adic algebraic groups |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2405.06094 |