{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:ULLRAONJ5RVZGP4UDWFBQ446VQ","short_pith_number":"pith:ULLRAONJ","schema_version":"1.0","canonical_sha256":"a2d71039a9ec6b933f941d8a18739eac07f99407785fc084a134b850cecf60d1","source":{"kind":"arxiv","id":"2108.03361","version":4},"attestation_state":"computed","paper":{"title":"Property (QT) for 3-manifold groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Hoang Thanh Nguyen, Suzhen Han, Wenyuan Yang","submitted_at":"2021-08-07T04:04:26Z","abstract_excerpt":"According to Bestvina-Bromberg-Fujiwara, a finitely generated group is said to have property (QT) if it acts isometrically on a finite product of quasi-trees so that orbital maps are quasi-isometric embeddings. We prove that the fundamental group $\\pi_1(M)$ of a compact, connected, orientable 3-manifold $M$ has property (QT) if and only if no summand in the sphere-disc decomposition of $M$ supports either Sol or Nil geometry. In particular, all compact, orientable, irreducible 3-manifold groups with nontrivial torus decomposition and not supporting Sol geometry have property (QT). In the cours"},"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":"2108.03361","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-08-07T04:04:26Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"9710624aebafbf5e4968ca9ee48fa0b31fd9c3288036d78c5ff8705a5d0caac8","abstract_canon_sha256":"84ac0a9d159ac92e1d290c781f8adbd60e2ece81c953ed46e8852425f8ebebd4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:28:08.961514Z","signature_b64":"v4Kr66uNADKfobKSEtklHZ0lz7C/Vsx4/B250oev6/itSnIcuU2MWzSbhKmlGDIQXOlywo2XwBBfmejUWjO0Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2d71039a9ec6b933f941d8a18739eac07f99407785fc084a134b850cecf60d1","last_reissued_at":"2026-07-05T10:28:08.961033Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:28:08.961033Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Property (QT) for 3-manifold groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Hoang Thanh Nguyen, Suzhen Han, Wenyuan Yang","submitted_at":"2021-08-07T04:04:26Z","abstract_excerpt":"According to Bestvina-Bromberg-Fujiwara, a finitely generated group is said to have property (QT) if it acts isometrically on a finite product of quasi-trees so that orbital maps are quasi-isometric embeddings. We prove that the fundamental group $\\pi_1(M)$ of a compact, connected, orientable 3-manifold $M$ has property (QT) if and only if no summand in the sphere-disc decomposition of $M$ supports either Sol or Nil geometry. In particular, all compact, orientable, irreducible 3-manifold groups with nontrivial torus decomposition and not supporting Sol geometry have property (QT). In the cours"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.03361","kind":"arxiv","version":4},"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/2108.03361/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":"2108.03361","created_at":"2026-07-05T10:28:08.961081+00:00"},{"alias_kind":"arxiv_version","alias_value":"2108.03361v4","created_at":"2026-07-05T10:28:08.961081+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.03361","created_at":"2026-07-05T10:28:08.961081+00:00"},{"alias_kind":"pith_short_12","alias_value":"ULLRAONJ5RVZ","created_at":"2026-07-05T10:28:08.961081+00:00"},{"alias_kind":"pith_short_16","alias_value":"ULLRAONJ5RVZGP4U","created_at":"2026-07-05T10:28:08.961081+00:00"},{"alias_kind":"pith_short_8","alias_value":"ULLRAONJ","created_at":"2026-07-05T10:28:08.961081+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2501.13600","citing_title":"Stable cylinders and fine structures for hyperbolic groups and curve graphs","ref_index":19,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ULLRAONJ5RVZGP4UDWFBQ446VQ","json":"https://pith.science/pith/ULLRAONJ5RVZGP4UDWFBQ446VQ.json","graph_json":"https://pith.science/api/pith-number/ULLRAONJ5RVZGP4UDWFBQ446VQ/graph.json","events_json":"https://pith.science/api/pith-number/ULLRAONJ5RVZGP4UDWFBQ446VQ/events.json","paper":"https://pith.science/paper/ULLRAONJ"},"agent_actions":{"view_html":"https://pith.science/pith/ULLRAONJ5RVZGP4UDWFBQ446VQ","download_json":"https://pith.science/pith/ULLRAONJ5RVZGP4UDWFBQ446VQ.json","view_paper":"https://pith.science/paper/ULLRAONJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2108.03361&json=true","fetch_graph":"https://pith.science/api/pith-number/ULLRAONJ5RVZGP4UDWFBQ446VQ/graph.json","fetch_events":"https://pith.science/api/pith-number/ULLRAONJ5RVZGP4UDWFBQ446VQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ULLRAONJ5RVZGP4UDWFBQ446VQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ULLRAONJ5RVZGP4UDWFBQ446VQ/action/storage_attestation","attest_author":"https://pith.science/pith/ULLRAONJ5RVZGP4UDWFBQ446VQ/action/author_attestation","sign_citation":"https://pith.science/pith/ULLRAONJ5RVZGP4UDWFBQ446VQ/action/citation_signature","submit_replication":"https://pith.science/pith/ULLRAONJ5RVZGP4UDWFBQ446VQ/action/replication_record"}},"created_at":"2026-07-05T10:28:08.961081+00:00","updated_at":"2026-07-05T10:28:08.961081+00:00"}