{"paper":{"title":"The linear $\\SL_2(\\Z)$-action on $\\T^n$: ergodic and von Neumann algebraic aspects","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Alain Valette, Paul Jolissaint","submitted_at":"2023-11-05T15:45:49Z","abstract_excerpt":"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 $\\T^n$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.02683","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2311.02683/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}