The linear $\SL_2(\Z)$-action on $\T^n$: ergodic and von Neumann algebraic aspects
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916199037665280 |
|---|---|
| author | Jolissaint, Paul Valette, Alain |
| author_facet | Jolissaint, Paul Valette, Alain |
| contents | The unique irreducible representation of $\SL_2(\R)$ on $\R^n$ induces an action, called the \textit{linear action}, of $\SL_2(\Z)$ on the torus $\T^n$ for every $n\geq 2$. For $n$ odd, it factors through $\PSL_2(\Z)$, so we denote by $G_n$ the group $\SL_2(\Z)$ for $n$ even, and $\PSL_2(\Z)$ for $n$ odd. We prove that the action is free and ergodic for every $n\geq 2$, that if $h\in \SL_2(\Z)$ is a hyperbolic element and if $n$ is even, then the action of the subgroup generated by $h$ is still ergodic, but also that, for $n$ odd, no amenable subgroup of $\PSL_2(\Z)$ acts ergodically on $\T^n$. We deduce also that every ergodic sub-equivalence relation $\Rr$ of the orbital equivalence relation $\mathcal{S}_n$ of $G_n$ on $\T^n$ is either amenable or rigid, extending a result by Ioana for $n=2$. This result has the following corollaries: firstly, for $n\geq 2$ even, if $H$ is a maximal amenable subgroup of $\SL_2(\Z)$ containing an hyperbolic matrix, then the associated crossed product II$_1$ factor $L^\infty(\T^n)\rtimes H$ is a maximal Haagerup subalgebra of $L^\infty(\T^n)\rtimes \SL_2(\Z)$; secondly , for every $n$, the fundamental group of $L^\infty(\T^n)\rtimes G_n$ is trivial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_02683 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The linear $\SL_2(\Z)$-action on $\T^n$: ergodic and von Neumann algebraic aspects Jolissaint, Paul Valette, Alain Operator Algebras 37A20 (Primary), 28D05, 46L36 (Secondary) The unique irreducible representation of $\SL_2(\R)$ on $\R^n$ induces an action, called the \textit{linear action}, of $\SL_2(\Z)$ on the torus $\T^n$ for every $n\geq 2$. For $n$ odd, it factors through $\PSL_2(\Z)$, so we denote by $G_n$ the group $\SL_2(\Z)$ for $n$ even, and $\PSL_2(\Z)$ for $n$ odd. We prove that the action is free and ergodic for every $n\geq 2$, that if $h\in \SL_2(\Z)$ is a hyperbolic element and if $n$ is even, then the action of the subgroup generated by $h$ is still ergodic, but also that, for $n$ odd, no amenable subgroup of $\PSL_2(\Z)$ acts ergodically on $\T^n$. We deduce also that every ergodic sub-equivalence relation $\Rr$ of the orbital equivalence relation $\mathcal{S}_n$ of $G_n$ on $\T^n$ is either amenable or rigid, extending a result by Ioana for $n=2$. This result has the following corollaries: firstly, for $n\geq 2$ even, if $H$ is a maximal amenable subgroup of $\SL_2(\Z)$ containing an hyperbolic matrix, then the associated crossed product II$_1$ factor $L^\infty(\T^n)\rtimes H$ is a maximal Haagerup subalgebra of $L^\infty(\T^n)\rtimes \SL_2(\Z)$; secondly , for every $n$, the fundamental group of $L^\infty(\T^n)\rtimes G_n$ is trivial. |
| title | The linear $\SL_2(\Z)$-action on $\T^n$: ergodic and von Neumann algebraic aspects |
| topic | Operator Algebras 37A20 (Primary), 28D05, 46L36 (Secondary) |
| url | https://arxiv.org/abs/2311.02683 |