{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:NQCK7MH7D4IRXMMWEB37H43B7D","short_pith_number":"pith:NQCK7MH7","canonical_record":{"source":{"id":"1806.00147","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-06-01T00:31:26Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"a719e05191657cb2fd40fbaceaa23a8060215ead8d410d94fc61994927b90cdf","abstract_canon_sha256":"c65b41f7a779d236265d1c9e6e64376dbe42ed305e4bf3a7dce599d111a00edb"},"schema_version":"1.0"},"canonical_sha256":"6c04afb0ff1f111bb1962077f3f361f8e08f3797c646b733dabfa5c140c8ce78","source":{"kind":"arxiv","id":"1806.00147","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.00147","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"arxiv_version","alias_value":"1806.00147v1","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00147","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"pith_short_12","alias_value":"NQCK7MH7D4IR","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NQCK7MH7D4IRXMMW","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NQCK7MH7","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:NQCK7MH7D4IRXMMWEB37H43B7D","target":"record","payload":{"canonical_record":{"source":{"id":"1806.00147","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-06-01T00:31:26Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"a719e05191657cb2fd40fbaceaa23a8060215ead8d410d94fc61994927b90cdf","abstract_canon_sha256":"c65b41f7a779d236265d1c9e6e64376dbe42ed305e4bf3a7dce599d111a00edb"},"schema_version":"1.0"},"canonical_sha256":"6c04afb0ff1f111bb1962077f3f361f8e08f3797c646b733dabfa5c140c8ce78","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:49.670175Z","signature_b64":"p+nJZWawQFHNnv5si9P9wroCuVbqdDHmZGKFlHWEuC6QNCNZsU+71a7OYN3JEtapmWFEaT4H+7SgcRpXr0/RCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c04afb0ff1f111bb1962077f3f361f8e08f3797c646b733dabfa5c140c8ce78","last_reissued_at":"2026-05-17T23:40:49.669384Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:49.669384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.00147","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"koQ/CJ5A+yCLajS+w+MsX+ltjePKfNUx/VVZNfd4qq+eS1VcEZNiWt/UlUoolKCH2E+A7akalxKnFpCROPSwBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T13:03:35.527161Z"},"content_sha256":"50e8b99b97792db11a2b6a55be8ba48231a2e68cad6aed3a01347732b8562b15","schema_version":"1.0","event_id":"sha256:50e8b99b97792db11a2b6a55be8ba48231a2e68cad6aed3a01347732b8562b15"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:NQCK7MH7D4IRXMMWEB37H43B7D","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.MG","authors_text":"Russell Ricks","submitted_at":"2018-06-01T00:31:26Z","abstract_excerpt":"We prove the following rank rigidity result for proper CAT(0) spaces with one-dimensional Tits boundaries: Let $\\Gamma$ be a group acting properly discontinuously, cocompactly, and by isometries on such a space $X$. If the Tits diameter of $\\partial X$ equals $\\pi$ and $\\Gamma$ does not act minimally on $\\partial X$, then $\\partial X$ is a spherical building or a spherical join. If $X$ is also geodesically complete, then $X$ is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of $\\par"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00147","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RxKj5AHgAuiDRMfwOkoNnnvl6ZKuvpyBkR0+J9zpHduQVKIk1REk1ARTa1kExeHj7De4f0LJLvLIBvJ3ZdStBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T13:03:35.527527Z"},"content_sha256":"28b149ddd2ad486ffa440119e8aed101abb21a3e8e8cd8d8902f2d4cebc30340","schema_version":"1.0","event_id":"sha256:28b149ddd2ad486ffa440119e8aed101abb21a3e8e8cd8d8902f2d4cebc30340"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NQCK7MH7D4IRXMMWEB37H43B7D/bundle.json","state_url":"https://pith.science/pith/NQCK7MH7D4IRXMMWEB37H43B7D/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NQCK7MH7D4IRXMMWEB37H43B7D/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-26T13:03:35Z","links":{"resolver":"https://pith.science/pith/NQCK7MH7D4IRXMMWEB37H43B7D","bundle":"https://pith.science/pith/NQCK7MH7D4IRXMMWEB37H43B7D/bundle.json","state":"https://pith.science/pith/NQCK7MH7D4IRXMMWEB37H43B7D/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NQCK7MH7D4IRXMMWEB37H43B7D/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NQCK7MH7D4IRXMMWEB37H43B7D","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":"c65b41f7a779d236265d1c9e6e64376dbe42ed305e4bf3a7dce599d111a00edb","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-06-01T00:31:26Z","title_canon_sha256":"a719e05191657cb2fd40fbaceaa23a8060215ead8d410d94fc61994927b90cdf"},"schema_version":"1.0","source":{"id":"1806.00147","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.00147","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"arxiv_version","alias_value":"1806.00147v1","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00147","created_at":"2026-05-17T23:40:49Z"},{"alias_kind":"pith_short_12","alias_value":"NQCK7MH7D4IR","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NQCK7MH7D4IRXMMW","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NQCK7MH7","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:28b149ddd2ad486ffa440119e8aed101abb21a3e8e8cd8d8902f2d4cebc30340","target":"graph","created_at":"2026-05-17T23:40:49Z","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 prove the following rank rigidity result for proper CAT(0) spaces with one-dimensional Tits boundaries: Let $\\Gamma$ be a group acting properly discontinuously, cocompactly, and by isometries on such a space $X$. If the Tits diameter of $\\partial X$ equals $\\pi$ and $\\Gamma$ does not act minimally on $\\partial X$, then $\\partial X$ is a spherical building or a spherical join. If $X$ is also geodesically complete, then $X$ is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of $\\par","authors_text":"Russell Ricks","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-06-01T00:31:26Z","title":"A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00147","kind":"arxiv","version":1},"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":"sha256:50e8b99b97792db11a2b6a55be8ba48231a2e68cad6aed3a01347732b8562b15","target":"record","created_at":"2026-05-17T23:40:49Z","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":"c65b41f7a779d236265d1c9e6e64376dbe42ed305e4bf3a7dce599d111a00edb","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-06-01T00:31:26Z","title_canon_sha256":"a719e05191657cb2fd40fbaceaa23a8060215ead8d410d94fc61994927b90cdf"},"schema_version":"1.0","source":{"id":"1806.00147","kind":"arxiv","version":1}},"canonical_sha256":"6c04afb0ff1f111bb1962077f3f361f8e08f3797c646b733dabfa5c140c8ce78","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6c04afb0ff1f111bb1962077f3f361f8e08f3797c646b733dabfa5c140c8ce78","first_computed_at":"2026-05-17T23:40:49.669384Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:49.669384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p+nJZWawQFHNnv5si9P9wroCuVbqdDHmZGKFlHWEuC6QNCNZsU+71a7OYN3JEtapmWFEaT4H+7SgcRpXr0/RCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:49.670175Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.00147","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50e8b99b97792db11a2b6a55be8ba48231a2e68cad6aed3a01347732b8562b15","sha256:28b149ddd2ad486ffa440119e8aed101abb21a3e8e8cd8d8902f2d4cebc30340"],"state_sha256":"c158b9a8ec738c730c9428617ae9824c253d002b2f8bd947df592b5d4a22b05a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LY0WYOqNsY/iNzuMbCZ3MPYig7t2Jn75K8xUXII9LpRkjxZtnxOS1DEI3N6ii+QduMR3Zzpr3OuRoSw3yO2dBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T13:03:35.529616Z","bundle_sha256":"6a340f5543cafbaec2eebf46ad052490836fe9b8a9b8d24e0880d1bf841276a6"}}