{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:PJN6C22DSDVTB6DNFZQ4EAO4CP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8049eda9f060f9c808317b9647f439bdf08b2ac9f77540deeb96fbca10679838","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-09-03T15:49:20Z","title_canon_sha256":"9ce4dd813b652863208ae9f56cd1d698b75293c6aa0566efb6e919292cd30853"},"schema_version":"1.0","source":{"id":"1809.00632","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.00632","created_at":"2026-05-17T23:52:58Z"},{"alias_kind":"arxiv_version","alias_value":"1809.00632v2","created_at":"2026-05-17T23:52:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.00632","created_at":"2026-05-17T23:52:58Z"},{"alias_kind":"pith_short_12","alias_value":"PJN6C22DSDVT","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"PJN6C22DSDVTB6DN","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"PJN6C22D","created_at":"2026-05-18T12:32:43Z"}],"graph_snapshots":[{"event_id":"sha256:60feb2ddc5563470e88b751d609966b20709988a65eb7c32f316d22740b40d85","target":"graph","created_at":"2026-05-17T23:52:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group $\\Gamma$ with respect to a sequence of groups $\\left\\{G_{n}\\right\\}_{n=1}^{\\infty}$, equipped with bi-invariant metrics $\\left\\{d_{n}\\right\\}_{n=1}^{\\infty}$. We consider the case $G_{n}=\\operatorname{U}\\left(n\\right)$ (resp. $G_{n}=\\operatorname{Sym}\\left(n\\right)$), equipped with the normalized Hilbert-Schmidt metric $d_{n}^{\\operatorname{HS}}$ (resp. the normalized Hamming metric $d_{n}^{\\operatorname{Hamming}}$). Our main result is that if $\\Gamma$ is infinite, hyperlinear (resp","authors_text":"Alexander Lubotzky, Oren Becker","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-09-03T15:49:20Z","title":"Group stability and Property (T)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00632","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d83e3faf305653babf38b46b5b2d1c61a6b4263aad77c83e1ee52f21b33b060e","target":"record","created_at":"2026-05-17T23:52:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8049eda9f060f9c808317b9647f439bdf08b2ac9f77540deeb96fbca10679838","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-09-03T15:49:20Z","title_canon_sha256":"9ce4dd813b652863208ae9f56cd1d698b75293c6aa0566efb6e919292cd30853"},"schema_version":"1.0","source":{"id":"1809.00632","kind":"arxiv","version":2}},"canonical_sha256":"7a5be16b4390eb30f86d2e61c201dc13d5936206a282b551cbb2c1ba6640fd87","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7a5be16b4390eb30f86d2e61c201dc13d5936206a282b551cbb2c1ba6640fd87","first_computed_at":"2026-05-17T23:52:58.841240Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:52:58.841240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UTA/yUmr0ixC5qQQ38T/YzaX3UnOxWv4SuQzpbesa2Jv2M1FoCXVplB23l5La410l+QB5xoZrmxthWWS4h17BQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:52:58.841858Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.00632","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d83e3faf305653babf38b46b5b2d1c61a6b4263aad77c83e1ee52f21b33b060e","sha256:60feb2ddc5563470e88b751d609966b20709988a65eb7c32f316d22740b40d85"],"state_sha256":"afec0d338376ddbd1ad4377aae66dc9b3a9933dfec827784131864f37a7f6598"}