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We prove that for any subgroup $G\\leq \\mathbb{R}^*_+$ which is realized as a fundamental group of a $\\rm II_1$ factor, there exists a $\\rm II_1$ factor $M$ which satisfies this property and whose fundamental group is $G$. Using this, we deduce that if $G,H \\leq \\mathbb{R}^*_+$ are realized as fundamental groups of $\\rm II_1$ factors (w"},"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":"1608.06426","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2016-08-23T09:14:53Z","cross_cats_sorted":[],"title_canon_sha256":"105c7bbcc8ae4861686e225ed79643767205e3df2215a554527bcba6db2bef25","abstract_canon_sha256":"99612841f306b04ab9d9b38babfda565fe5921dec6f7f01d319edb1035e17fd1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:57.043384Z","signature_b64":"5H5YmLEs8jLFIM9FdRYfXb18ydPqF4V7wq4zqqa75BSxfyu8YUCEPuWYOpJQgxAv9xB0JPL0BHzSM6mU8cz7Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c1e6aaca5ede72fd299349bc8e7832145e3b23f949837241ee8022122756e556","last_reissued_at":"2026-05-17T23:54:57.042782Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:57.042782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On fundamental groups of tensor product $\\rm II_1$ factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Yusuke Isono","submitted_at":"2016-08-23T09:14:53Z","abstract_excerpt":"Let $M$ be a $\\rm II_1$ factor and let $\\mathcal{F}(M)$ denote the fundamental group of $M$. In this article, we study the following property of $M$: for arbitrary $\\rm II_1$ factor $B$, we have $\\mathcal{F}(M \\overline{\\otimes} B)=\\mathcal{F}(M)\\mathcal{F}(B)$. We prove that for any subgroup $G\\leq \\mathbb{R}^*_+$ which is realized as a fundamental group of a $\\rm II_1$ factor, there exists a $\\rm II_1$ factor $M$ which satisfies this property and whose fundamental group is $G$. Using this, we deduce that if $G,H \\leq \\mathbb{R}^*_+$ are realized as fundamental groups of $\\rm II_1$ factors (w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.06426","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1608.06426","created_at":"2026-05-17T23:54:57.042863+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.06426v2","created_at":"2026-05-17T23:54:57.042863+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.06426","created_at":"2026-05-17T23:54:57.042863+00:00"},{"alias_kind":"pith_short_12","alias_value":"YHTKVSS63ZZP","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_16","alias_value":"YHTKVSS63ZZP2KMT","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_8","alias_value":"YHTKVSS6","created_at":"2026-05-18T12:30:53.716459+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YHTKVSS63ZZP2KMTJG6I46BSCR","json":"https://pith.science/pith/YHTKVSS63ZZP2KMTJG6I46BSCR.json","graph_json":"https://pith.science/api/pith-number/YHTKVSS63ZZP2KMTJG6I46BSCR/graph.json","events_json":"https://pith.science/api/pith-number/YHTKVSS63ZZP2KMTJG6I46BSCR/events.json","paper":"https://pith.science/paper/YHTKVSS6"},"agent_actions":{"view_html":"https://pith.science/pith/YHTKVSS63ZZP2KMTJG6I46BSCR","download_json":"https://pith.science/pith/YHTKVSS63ZZP2KMTJG6I46BSCR.json","view_paper":"https://pith.science/paper/YHTKVSS6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.06426&json=true","fetch_graph":"https://pith.science/api/pith-number/YHTKVSS63ZZP2KMTJG6I46BSCR/graph.json","fetch_events":"https://pith.science/api/pith-number/YHTKVSS63ZZP2KMTJG6I46BSCR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YHTKVSS63ZZP2KMTJG6I46BSCR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YHTKVSS63ZZP2KMTJG6I46BSCR/action/storage_attestation","attest_author":"https://pith.science/pith/YHTKVSS63ZZP2KMTJG6I46BSCR/action/author_attestation","sign_citation":"https://pith.science/pith/YHTKVSS63ZZP2KMTJG6I46BSCR/action/citation_signature","submit_replication":"https://pith.science/pith/YHTKVSS63ZZP2KMTJG6I46BSCR/action/replication_record"}},"created_at":"2026-05-17T23:54:57.042863+00:00","updated_at":"2026-05-17T23:54:57.042863+00:00"}