{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:4JRRISVW4ZLQYXNQJDLM2EOGX7","short_pith_number":"pith:4JRRISVW","schema_version":"1.0","canonical_sha256":"e263144ab6e6570c5db048d6cd11c6bfd874390039279fc353af5b91b461d0b3","source":{"kind":"arxiv","id":"2311.02683","version":3},"attestation_state":"computed","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 $\\T^n$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2311.02683","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OA","submitted_at":"2023-11-05T15:45:49Z","cross_cats_sorted":[],"title_canon_sha256":"ef99db2e15facd267bdc5916e3bc2ef255ab7da093b9b8682b5e4a01f52e1113","abstract_canon_sha256":"e473de254982342e64a66b960f554472492c1f9266698461325426b257d88459"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:06:11.835856Z","signature_b64":"M7hwclDMJ+/zjHnF1H+3/PquUHjDe6hVdI/dY8UmCjGl6dHgV9YfQepwjMv5QRvlM3Tv9d7ZBpZsM5SC/9ZvDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e263144ab6e6570c5db048d6cd11c6bfd874390039279fc353af5b91b461d0b3","last_reissued_at":"2026-07-05T08:06:11.835392Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:06:11.835392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2311.02683","created_at":"2026-07-05T08:06:11.835449+00:00"},{"alias_kind":"arxiv_version","alias_value":"2311.02683v3","created_at":"2026-07-05T08:06:11.835449+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.02683","created_at":"2026-07-05T08:06:11.835449+00:00"},{"alias_kind":"pith_short_12","alias_value":"4JRRISVW4ZLQ","created_at":"2026-07-05T08:06:11.835449+00:00"},{"alias_kind":"pith_short_16","alias_value":"4JRRISVW4ZLQYXNQ","created_at":"2026-07-05T08:06:11.835449+00:00"},{"alias_kind":"pith_short_8","alias_value":"4JRRISVW","created_at":"2026-07-05T08:06:11.835449+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.14358","citing_title":"Spaces of UCP maps and subalgebras of von Neumann algebras","ref_index":22,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4JRRISVW4ZLQYXNQJDLM2EOGX7","json":"https://pith.science/pith/4JRRISVW4ZLQYXNQJDLM2EOGX7.json","graph_json":"https://pith.science/api/pith-number/4JRRISVW4ZLQYXNQJDLM2EOGX7/graph.json","events_json":"https://pith.science/api/pith-number/4JRRISVW4ZLQYXNQJDLM2EOGX7/events.json","paper":"https://pith.science/paper/4JRRISVW"},"agent_actions":{"view_html":"https://pith.science/pith/4JRRISVW4ZLQYXNQJDLM2EOGX7","download_json":"https://pith.science/pith/4JRRISVW4ZLQYXNQJDLM2EOGX7.json","view_paper":"https://pith.science/paper/4JRRISVW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2311.02683&json=true","fetch_graph":"https://pith.science/api/pith-number/4JRRISVW4ZLQYXNQJDLM2EOGX7/graph.json","fetch_events":"https://pith.science/api/pith-number/4JRRISVW4ZLQYXNQJDLM2EOGX7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4JRRISVW4ZLQYXNQJDLM2EOGX7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4JRRISVW4ZLQYXNQJDLM2EOGX7/action/storage_attestation","attest_author":"https://pith.science/pith/4JRRISVW4ZLQYXNQJDLM2EOGX7/action/author_attestation","sign_citation":"https://pith.science/pith/4JRRISVW4ZLQYXNQJDLM2EOGX7/action/citation_signature","submit_replication":"https://pith.science/pith/4JRRISVW4ZLQYXNQJDLM2EOGX7/action/replication_record"}},"created_at":"2026-07-05T08:06:11.835449+00:00","updated_at":"2026-07-05T08:06:11.835449+00:00"}