{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:PQL3ADMKVQABLHEEMOFTOKZWBC","short_pith_number":"pith:PQL3ADMK","canonical_record":{"source":{"id":"2208.04778","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2022-08-09T13:47:31Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"78a955bae6711981adcad8265a93d93eb49853c2f6cdf93acaad55ac7608e036","abstract_canon_sha256":"273b68100c055d41689daa52473116cbfe3530b5eba2891b51cb94e91630d942"},"schema_version":"1.0"},"canonical_sha256":"7c17b00d8aac00159c84638b372b360891187807c97644bc3276de9bb13e45e5","source":{"kind":"arxiv","id":"2208.04778","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2208.04778","created_at":"2026-07-05T04:47:00Z"},{"alias_kind":"arxiv_version","alias_value":"2208.04778v1","created_at":"2026-07-05T04:47:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2208.04778","created_at":"2026-07-05T04:47:00Z"},{"alias_kind":"pith_short_12","alias_value":"PQL3ADMKVQAB","created_at":"2026-07-05T04:47:00Z"},{"alias_kind":"pith_short_16","alias_value":"PQL3ADMKVQABLHEE","created_at":"2026-07-05T04:47:00Z"},{"alias_kind":"pith_short_8","alias_value":"PQL3ADMK","created_at":"2026-07-05T04:47:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:PQL3ADMKVQABLHEEMOFTOKZWBC","target":"record","payload":{"canonical_record":{"source":{"id":"2208.04778","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2022-08-09T13:47:31Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"78a955bae6711981adcad8265a93d93eb49853c2f6cdf93acaad55ac7608e036","abstract_canon_sha256":"273b68100c055d41689daa52473116cbfe3530b5eba2891b51cb94e91630d942"},"schema_version":"1.0"},"canonical_sha256":"7c17b00d8aac00159c84638b372b360891187807c97644bc3276de9bb13e45e5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:47:00.931628Z","signature_b64":"sjksJKQy4p+Pc+zf4BJCMDeRCTyDHPxqBbJFOoRFJbyVFx35/Psq50pqnJbdrfElGKTS2sz8+Jjrq9N7nQnHCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c17b00d8aac00159c84638b372b360891187807c97644bc3276de9bb13e45e5","last_reissued_at":"2026-07-05T04:47:00.931242Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:47:00.931242Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2208.04778","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":"2026-07-05T04:47:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j6AW2bADZDIJPz1ouWphNqUVZv2w5ECujVs2w/TI5d16bMCLtwbk6dglTfdREtZG2qB20eCxCBLTFanSd70sBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T20:42:13.514219Z"},"content_sha256":"8ba0e6bdbc949e89bde1881cac9d88fffb1f11d873fe45bab8ae99bf0991df57","schema_version":"1.0","event_id":"sha256:8ba0e6bdbc949e89bde1881cac9d88fffb1f11d873fe45bab8ae99bf0991df57"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:PQL3ADMKVQABLHEEMOFTOKZWBC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Genericity of sublinearly Morse directions in CAT(0) spaces and the Teichm\\\"uller space","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.GR","authors_text":"Ilya Gekhtman, Kasra Rafi, Yulan Qing","submitted_at":"2022-08-09T13:47:31Z","abstract_excerpt":"We show that the sublinearly Morse directions in the visual boundary of a rank-1 CAT(0) space with a geometric group action are generic in several commonly studied senses of the word, namely with respect to Patterson-Sullivan measures and stationary measures for random walks. We deduce that the sublinearly Morse boundary is a model of the Poisson boundary for finitely supported random walks on groups acting geometrically on rank-1 CAT (0) spaces. We prove an analogous result for mapping class group actions on Teichm\\\"uller space. Our main technical tool is a criterion, valid in any unique geod"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.04778","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2208.04778/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:47:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yDbRao02AdNQ7Xwl76IprBMbKR3TLkQnsA+WoFPD/UKxE4sVN1g6lSRqdAtgAJKTcrhvlm9uB60I3xi07JSKDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T20:42:13.515548Z"},"content_sha256":"b2a91052106a9ef13240212bc9efd01757b5aa3d5176266bb7201e5a02312392","schema_version":"1.0","event_id":"sha256:b2a91052106a9ef13240212bc9efd01757b5aa3d5176266bb7201e5a02312392"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PQL3ADMKVQABLHEEMOFTOKZWBC/bundle.json","state_url":"https://pith.science/pith/PQL3ADMKVQABLHEEMOFTOKZWBC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PQL3ADMKVQABLHEEMOFTOKZWBC/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":"2026-08-21T20:42:13Z","links":{"resolver":"https://pith.science/pith/PQL3ADMKVQABLHEEMOFTOKZWBC","bundle":"https://pith.science/pith/PQL3ADMKVQABLHEEMOFTOKZWBC/bundle.json","state":"https://pith.science/pith/PQL3ADMKVQABLHEEMOFTOKZWBC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PQL3ADMKVQABLHEEMOFTOKZWBC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:PQL3ADMKVQABLHEEMOFTOKZWBC","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":"273b68100c055d41689daa52473116cbfe3530b5eba2891b51cb94e91630d942","cross_cats_sorted":["math.DS"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2022-08-09T13:47:31Z","title_canon_sha256":"78a955bae6711981adcad8265a93d93eb49853c2f6cdf93acaad55ac7608e036"},"schema_version":"1.0","source":{"id":"2208.04778","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2208.04778","created_at":"2026-07-05T04:47:00Z"},{"alias_kind":"arxiv_version","alias_value":"2208.04778v1","created_at":"2026-07-05T04:47:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2208.04778","created_at":"2026-07-05T04:47:00Z"},{"alias_kind":"pith_short_12","alias_value":"PQL3ADMKVQAB","created_at":"2026-07-05T04:47:00Z"},{"alias_kind":"pith_short_16","alias_value":"PQL3ADMKVQABLHEE","created_at":"2026-07-05T04:47:00Z"},{"alias_kind":"pith_short_8","alias_value":"PQL3ADMK","created_at":"2026-07-05T04:47:00Z"}],"graph_snapshots":[{"event_id":"sha256:b2a91052106a9ef13240212bc9efd01757b5aa3d5176266bb7201e5a02312392","target":"graph","created_at":"2026-07-05T04:47:00Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2208.04778/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that the sublinearly Morse directions in the visual boundary of a rank-1 CAT(0) space with a geometric group action are generic in several commonly studied senses of the word, namely with respect to Patterson-Sullivan measures and stationary measures for random walks. We deduce that the sublinearly Morse boundary is a model of the Poisson boundary for finitely supported random walks on groups acting geometrically on rank-1 CAT (0) spaces. We prove an analogous result for mapping class group actions on Teichm\\\"uller space. Our main technical tool is a criterion, valid in any unique geod","authors_text":"Ilya Gekhtman, Kasra Rafi, Yulan Qing","cross_cats":["math.DS"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2022-08-09T13:47:31Z","title":"Genericity of sublinearly Morse directions in CAT(0) spaces and the Teichm\\\"uller space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.04778","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":"sha256:8ba0e6bdbc949e89bde1881cac9d88fffb1f11d873fe45bab8ae99bf0991df57","target":"record","created_at":"2026-07-05T04:47:00Z","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":"273b68100c055d41689daa52473116cbfe3530b5eba2891b51cb94e91630d942","cross_cats_sorted":["math.DS"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2022-08-09T13:47:31Z","title_canon_sha256":"78a955bae6711981adcad8265a93d93eb49853c2f6cdf93acaad55ac7608e036"},"schema_version":"1.0","source":{"id":"2208.04778","kind":"arxiv","version":1}},"canonical_sha256":"7c17b00d8aac00159c84638b372b360891187807c97644bc3276de9bb13e45e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7c17b00d8aac00159c84638b372b360891187807c97644bc3276de9bb13e45e5","first_computed_at":"2026-07-05T04:47:00.931242Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:47:00.931242Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sjksJKQy4p+Pc+zf4BJCMDeRCTyDHPxqBbJFOoRFJbyVFx35/Psq50pqnJbdrfElGKTS2sz8+Jjrq9N7nQnHCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T04:47:00.931628Z","signed_message":"canonical_sha256_bytes"},"source_id":"2208.04778","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8ba0e6bdbc949e89bde1881cac9d88fffb1f11d873fe45bab8ae99bf0991df57","sha256:b2a91052106a9ef13240212bc9efd01757b5aa3d5176266bb7201e5a02312392"],"state_sha256":"3f286f9132382f8555d68ed0562a8411fd095c8f9fea2f16b90d46b0dde01429"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TK7wwj3x2aUBfunMMwJxD4KBFrzog+1cH1fpRI3mXyWw6ulfoVvvtHN1DpLseuvT/J3U4spE8BlR+Y5Rw/CBCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T20:42:13.523121Z","bundle_sha256":"c527b2de9fee49149f7024a626020be7031769318ebc484b02b7df85ccf30741"}}