{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:M47Q5MCXFAFMZIHD2RZ6ZDRRKA","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":"4f9712b231227a87578aec3b2214edacbafbee8586125b7f01fe3cdc515b69fb","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-06-12T22:52:17Z","title_canon_sha256":"394f0e319c83fd086b8eb1bb550430a1bbd1132abbec84f9d338bfccb1e4b70c"},"schema_version":"1.0","source":{"id":"1906.05417","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.05417","created_at":"2026-05-17T23:42:41Z"},{"alias_kind":"arxiv_version","alias_value":"1906.05417v2","created_at":"2026-05-17T23:42:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.05417","created_at":"2026-05-17T23:42:41Z"},{"alias_kind":"pith_short_12","alias_value":"M47Q5MCXFAFM","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_16","alias_value":"M47Q5MCXFAFMZIHD","created_at":"2026-05-18T12:33:21Z"},{"alias_kind":"pith_short_8","alias_value":"M47Q5MCX","created_at":"2026-05-18T12:33:21Z"}],"graph_snapshots":[{"event_id":"sha256:0caf5ad2a84c463e846a8107dbbc96d8a7665df5afa9e0593caaf952c1960e6a","target":"graph","created_at":"2026-05-17T23:42:41Z","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":"We consider two random group models: the hexagonal model and the square model, defined as the quotient of a free group by a random set of reduced words of length four and six respectively. Our first main result is that in this model there exists a sharp density threshold for Kazhdan's Property (T) and it equals 1/3. Our second main result is that for densities < 3/8 a random group in the square model with overwhelming probability does not have Property (T). Moreover, we provide a new version of the Isoperimetric Inequality that concerns non-planar diagrams and we introduce new geometrical tool","authors_text":"Tomasz Odrzyg\\'o\\'zd\\'z","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-06-12T22:52:17Z","title":"Bent walls for random groups in the square and hexagonal model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.05417","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":"sha256:2a7af7c70708d9d88b14ef80ad89f571955bac77b4daa4d01be60b1d56d1fd75","target":"record","created_at":"2026-05-17T23:42:41Z","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":"4f9712b231227a87578aec3b2214edacbafbee8586125b7f01fe3cdc515b69fb","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-06-12T22:52:17Z","title_canon_sha256":"394f0e319c83fd086b8eb1bb550430a1bbd1132abbec84f9d338bfccb1e4b70c"},"schema_version":"1.0","source":{"id":"1906.05417","kind":"arxiv","version":2}},"canonical_sha256":"673f0eb057280acca0e3d473ec8e315023d40933ed82e9a0de22d0e30bdf4857","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"673f0eb057280acca0e3d473ec8e315023d40933ed82e9a0de22d0e30bdf4857","first_computed_at":"2026-05-17T23:42:41.374799Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:41.374799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YTpk4Nwdc3fBFDIOObxibzdd0AsLyIFPiRK4ce6pVAL7Hfxzq176G4a6BhPoiI6Vnv6kBXk8WBtAdek4F7KTCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:41.375354Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.05417","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a7af7c70708d9d88b14ef80ad89f571955bac77b4daa4d01be60b1d56d1fd75","sha256:0caf5ad2a84c463e846a8107dbbc96d8a7665df5afa9e0593caaf952c1960e6a"],"state_sha256":"4cf2caae069c8316dca08a1b0bd714d21aa92e0d9bb02bbd6848aa425c3ea0d5"}