{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:T3EA5DL2RNEUAECWJS5ZHWMIIL","short_pith_number":"pith:T3EA5DL2","canonical_record":{"source":{"id":"2004.14624","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2020-04-30T07:55:11Z","cross_cats_sorted":["math.MG","math.PR"],"title_canon_sha256":"09ad5bdbd20a04f63acf2da1776c0e808d7c987f90b2b313aa7e057cb51b4187","abstract_canon_sha256":"108408bd287c29f7a68395094c43cd6f02086ae671091514c65ef842daa4fe4b"},"schema_version":"1.0"},"canonical_sha256":"9ec80e8d7a8b494010564cbb93d98842f66f91409b9f13f426b6ceba178c0205","source":{"kind":"arxiv","id":"2004.14624","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2004.14624","created_at":"2026-07-05T04:54:43Z"},{"alias_kind":"arxiv_version","alias_value":"2004.14624v2","created_at":"2026-07-05T04:54:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2004.14624","created_at":"2026-07-05T04:54:43Z"},{"alias_kind":"pith_short_12","alias_value":"T3EA5DL2RNEU","created_at":"2026-07-05T04:54:43Z"},{"alias_kind":"pith_short_16","alias_value":"T3EA5DL2RNEUAECW","created_at":"2026-07-05T04:54:43Z"},{"alias_kind":"pith_short_8","alias_value":"T3EA5DL2","created_at":"2026-07-05T04:54:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:T3EA5DL2RNEUAECWJS5ZHWMIIL","target":"record","payload":{"canonical_record":{"source":{"id":"2004.14624","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2020-04-30T07:55:11Z","cross_cats_sorted":["math.MG","math.PR"],"title_canon_sha256":"09ad5bdbd20a04f63acf2da1776c0e808d7c987f90b2b313aa7e057cb51b4187","abstract_canon_sha256":"108408bd287c29f7a68395094c43cd6f02086ae671091514c65ef842daa4fe4b"},"schema_version":"1.0"},"canonical_sha256":"9ec80e8d7a8b494010564cbb93d98842f66f91409b9f13f426b6ceba178c0205","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:54:43.783768Z","signature_b64":"x2ZnQ+Tt2TisnRfa3UOyqrYWY1f3kDhT4vy9AsfB7a+aJLYhlSaJ4virjcQjAYlpDeFijH+ckQgWzNlTIq0YCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ec80e8d7a8b494010564cbb93d98842f66f91409b9f13f426b6ceba178c0205","last_reissued_at":"2026-07-05T04:54:43.783425Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:54:43.783425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2004.14624","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:54:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/TIbRZ3YdRSOcoTJ/1p4wdYqNh2aIMymgSd7Bt75Oab2bGJjO2Jn5hOALB0hsBFLoN0vO831QDG79c6GXzzTDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T13:18:16.907015Z"},"content_sha256":"fa18fdd747fc8964eff5b11e6dbadf333bf46d90fcc8c1fc5c95554beecf1d7a","schema_version":"1.0","event_id":"sha256:fa18fdd747fc8964eff5b11e6dbadf333bf46d90fcc8c1fc5c95554beecf1d7a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:T3EA5DL2RNEUAECWJS5ZHWMIIL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An embedding of the Morse boundary in the Martin boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.PR"],"primary_cat":"math.GR","authors_text":"Ilya Gekhtman, Matthew Cordes (D-MATH), Matthieu Dussaule (LMJL)","submitted_at":"2020-04-30T07:55:11Z","abstract_excerpt":"We construct a one-to-one continuous map from the Morse boundary of a hierarchically hyperbolic group to its Martin boundary. This construction is based on deviation inequalities generalizing Ancona's work on hyperbolic groups. This provides a possibly new metrizable topology on the Morse boundary of such groups. We also prove that the Morse boundary has measure 0 with respect to the harmonic measure unless the group is hyperbolic."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2004.14624","kind":"arxiv","version":2},"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/2004.14624/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":"2026-07-05T04:54:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cwVaVDQPQ+ETyCKdTYmGVlYueoFbNTgwynRCUgddpRNOA9/taYQdQaFVcX2b8Tv/6aFHWIClR9WIBrlT/XLfCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T13:18:16.907466Z"},"content_sha256":"cf82e059aaaf0f8aacd9bd889eefac5b11d8c6ff0e1315988aa98d276a38e6df","schema_version":"1.0","event_id":"sha256:cf82e059aaaf0f8aacd9bd889eefac5b11d8c6ff0e1315988aa98d276a38e6df"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/T3EA5DL2RNEUAECWJS5ZHWMIIL/bundle.json","state_url":"https://pith.science/pith/T3EA5DL2RNEUAECWJS5ZHWMIIL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/T3EA5DL2RNEUAECWJS5ZHWMIIL/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":"2026-08-17T13:18:16Z","links":{"resolver":"https://pith.science/pith/T3EA5DL2RNEUAECWJS5ZHWMIIL","bundle":"https://pith.science/pith/T3EA5DL2RNEUAECWJS5ZHWMIIL/bundle.json","state":"https://pith.science/pith/T3EA5DL2RNEUAECWJS5ZHWMIIL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/T3EA5DL2RNEUAECWJS5ZHWMIIL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:T3EA5DL2RNEUAECWJS5ZHWMIIL","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":"108408bd287c29f7a68395094c43cd6f02086ae671091514c65ef842daa4fe4b","cross_cats_sorted":["math.MG","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2020-04-30T07:55:11Z","title_canon_sha256":"09ad5bdbd20a04f63acf2da1776c0e808d7c987f90b2b313aa7e057cb51b4187"},"schema_version":"1.0","source":{"id":"2004.14624","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2004.14624","created_at":"2026-07-05T04:54:43Z"},{"alias_kind":"arxiv_version","alias_value":"2004.14624v2","created_at":"2026-07-05T04:54:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2004.14624","created_at":"2026-07-05T04:54:43Z"},{"alias_kind":"pith_short_12","alias_value":"T3EA5DL2RNEU","created_at":"2026-07-05T04:54:43Z"},{"alias_kind":"pith_short_16","alias_value":"T3EA5DL2RNEUAECW","created_at":"2026-07-05T04:54:43Z"},{"alias_kind":"pith_short_8","alias_value":"T3EA5DL2","created_at":"2026-07-05T04:54:43Z"}],"graph_snapshots":[{"event_id":"sha256:cf82e059aaaf0f8aacd9bd889eefac5b11d8c6ff0e1315988aa98d276a38e6df","target":"graph","created_at":"2026-07-05T04:54:43Z","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/2004.14624/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We construct a one-to-one continuous map from the Morse boundary of a hierarchically hyperbolic group to its Martin boundary. This construction is based on deviation inequalities generalizing Ancona's work on hyperbolic groups. This provides a possibly new metrizable topology on the Morse boundary of such groups. We also prove that the Morse boundary has measure 0 with respect to the harmonic measure unless the group is hyperbolic.","authors_text":"Ilya Gekhtman, Matthew Cordes (D-MATH), Matthieu Dussaule (LMJL)","cross_cats":["math.MG","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2020-04-30T07:55:11Z","title":"An embedding of the Morse boundary in the Martin boundary"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2004.14624","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:fa18fdd747fc8964eff5b11e6dbadf333bf46d90fcc8c1fc5c95554beecf1d7a","target":"record","created_at":"2026-07-05T04:54:43Z","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":"108408bd287c29f7a68395094c43cd6f02086ae671091514c65ef842daa4fe4b","cross_cats_sorted":["math.MG","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2020-04-30T07:55:11Z","title_canon_sha256":"09ad5bdbd20a04f63acf2da1776c0e808d7c987f90b2b313aa7e057cb51b4187"},"schema_version":"1.0","source":{"id":"2004.14624","kind":"arxiv","version":2}},"canonical_sha256":"9ec80e8d7a8b494010564cbb93d98842f66f91409b9f13f426b6ceba178c0205","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9ec80e8d7a8b494010564cbb93d98842f66f91409b9f13f426b6ceba178c0205","first_computed_at":"2026-07-05T04:54:43.783425Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:54:43.783425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x2ZnQ+Tt2TisnRfa3UOyqrYWY1f3kDhT4vy9AsfB7a+aJLYhlSaJ4virjcQjAYlpDeFijH+ckQgWzNlTIq0YCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T04:54:43.783768Z","signed_message":"canonical_sha256_bytes"},"source_id":"2004.14624","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa18fdd747fc8964eff5b11e6dbadf333bf46d90fcc8c1fc5c95554beecf1d7a","sha256:cf82e059aaaf0f8aacd9bd889eefac5b11d8c6ff0e1315988aa98d276a38e6df"],"state_sha256":"4e708909d3ee2dc8948c06b85f80a3b28378a1eabc681562efdc6a563dc644be"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rkGFDCNKgLybnzNw4YprnMgYXQjBqtYDL+srDqclxqDzRGooqY159bcOeI8dPlr5qbqVg76aL0s7LPcJhvJjBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T13:18:16.910609Z","bundle_sha256":"6e2601571f72e6b46e46fe0801651062074b129ce5a417b26f97cdeff86bce1d"}}