{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:2XSCLK6MNPDKRPQSUL355M7LOU","short_pith_number":"pith:2XSCLK6M","canonical_record":{"source":{"id":"2401.04635","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-01-09T16:05:32Z","cross_cats_sorted":["math.DS","math.GT","math.OA"],"title_canon_sha256":"e0ca828ae060e3f9de7098ad2386adfd0c98e317b8d01f58ee3e8a3d90c187bd","abstract_canon_sha256":"380323c8d9cc027e49e839a41d86b2baab27bd7257409118203e34318dd627a6"},"schema_version":"1.0"},"canonical_sha256":"d5e425abcc6bc6a8be12a2f7deb3eb75043f5cb3dd595d2d3870c7b252033be5","source":{"kind":"arxiv","id":"2401.04635","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.04635","created_at":"2026-07-05T07:31:47Z"},{"alias_kind":"arxiv_version","alias_value":"2401.04635v1","created_at":"2026-07-05T07:31:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.04635","created_at":"2026-07-05T07:31:47Z"},{"alias_kind":"pith_short_12","alias_value":"2XSCLK6MNPDK","created_at":"2026-07-05T07:31:47Z"},{"alias_kind":"pith_short_16","alias_value":"2XSCLK6MNPDKRPQS","created_at":"2026-07-05T07:31:47Z"},{"alias_kind":"pith_short_8","alias_value":"2XSCLK6M","created_at":"2026-07-05T07:31:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:2XSCLK6MNPDKRPQSUL355M7LOU","target":"record","payload":{"canonical_record":{"source":{"id":"2401.04635","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-01-09T16:05:32Z","cross_cats_sorted":["math.DS","math.GT","math.OA"],"title_canon_sha256":"e0ca828ae060e3f9de7098ad2386adfd0c98e317b8d01f58ee3e8a3d90c187bd","abstract_canon_sha256":"380323c8d9cc027e49e839a41d86b2baab27bd7257409118203e34318dd627a6"},"schema_version":"1.0"},"canonical_sha256":"d5e425abcc6bc6a8be12a2f7deb3eb75043f5cb3dd595d2d3870c7b252033be5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:31:47.445748Z","signature_b64":"puhNx/l1MyN/pEVJMhOMZJj+OuBR15qMzEOWpOxLeeSxwJXnSkJ9lA/+3JP7EKat1iCgu4h5CKx4Tr6tFbxACQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5e425abcc6bc6a8be12a2f7deb3eb75043f5cb3dd595d2d3870c7b252033be5","last_reissued_at":"2026-07-05T07:31:47.445395Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:31:47.445395Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2401.04635","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":"2026-07-05T07:31:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zBIpvJlAHhY52O14dToXeWdU3QQQeoHvyRuHMuDWRPIaBdxitVTsJwmpXhMuANRnzqCsgrjf+wV941VfzDAKAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T16:54:04.641808Z"},"content_sha256":"4769f70daf93911276355c95070bab7350a7bb406e903efcaef2377762e74711","schema_version":"1.0","event_id":"sha256:4769f70daf93911276355c95070bab7350a7bb406e903efcaef2377762e74711"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:2XSCLK6MNPDKRPQSUL355M7LOU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Graph products and measure equivalence: classification, rigidity, and quantitative aspects","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS","math.GT","math.OA"],"primary_cat":"math.GR","authors_text":"Amandine Escalier, Camille Horbez","submitted_at":"2024-01-09T16:05:32Z","abstract_excerpt":"We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no partial conjugation. This finds applications to their classification up to commensurability, and up to isomorphism, and to the study of their automorphism groups. We also derive structural properties of von Neumann algebras associated to probability measure-preserving actions of graph products. Variations of the measure equivalence classification statement "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.04635","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/2401.04635/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":"2026-07-05T07:31:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SaI/JjhlE6EvGa2luJHHQFiCmh6sXWRDrAXku01A2CjTMs9OBk50L/TzYUa++FtIy65oMV7KobWw6hR2CrSuAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T16:54:04.642384Z"},"content_sha256":"ef0609407ad037ffc6491788271ba9604b799b92492f10ccb60a353dd59b13e1","schema_version":"1.0","event_id":"sha256:ef0609407ad037ffc6491788271ba9604b799b92492f10ccb60a353dd59b13e1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2XSCLK6MNPDKRPQSUL355M7LOU/bundle.json","state_url":"https://pith.science/pith/2XSCLK6MNPDKRPQSUL355M7LOU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2XSCLK6MNPDKRPQSUL355M7LOU/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":"2026-08-09T16:54:04Z","links":{"resolver":"https://pith.science/pith/2XSCLK6MNPDKRPQSUL355M7LOU","bundle":"https://pith.science/pith/2XSCLK6MNPDKRPQSUL355M7LOU/bundle.json","state":"https://pith.science/pith/2XSCLK6MNPDKRPQSUL355M7LOU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2XSCLK6MNPDKRPQSUL355M7LOU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:2XSCLK6MNPDKRPQSUL355M7LOU","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":"380323c8d9cc027e49e839a41d86b2baab27bd7257409118203e34318dd627a6","cross_cats_sorted":["math.DS","math.GT","math.OA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-01-09T16:05:32Z","title_canon_sha256":"e0ca828ae060e3f9de7098ad2386adfd0c98e317b8d01f58ee3e8a3d90c187bd"},"schema_version":"1.0","source":{"id":"2401.04635","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.04635","created_at":"2026-07-05T07:31:47Z"},{"alias_kind":"arxiv_version","alias_value":"2401.04635v1","created_at":"2026-07-05T07:31:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.04635","created_at":"2026-07-05T07:31:47Z"},{"alias_kind":"pith_short_12","alias_value":"2XSCLK6MNPDK","created_at":"2026-07-05T07:31:47Z"},{"alias_kind":"pith_short_16","alias_value":"2XSCLK6MNPDKRPQS","created_at":"2026-07-05T07:31:47Z"},{"alias_kind":"pith_short_8","alias_value":"2XSCLK6M","created_at":"2026-07-05T07:31:47Z"}],"graph_snapshots":[{"event_id":"sha256:ef0609407ad037ffc6491788271ba9604b799b92492f10ccb60a353dd59b13e1","target":"graph","created_at":"2026-07-05T07:31:47Z","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/2401.04635/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no partial conjugation. This finds applications to their classification up to commensurability, and up to isomorphism, and to the study of their automorphism groups. We also derive structural properties of von Neumann algebras associated to probability measure-preserving actions of graph products. Variations of the measure equivalence classification statement ","authors_text":"Amandine Escalier, Camille Horbez","cross_cats":["math.DS","math.GT","math.OA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-01-09T16:05:32Z","title":"Graph products and measure equivalence: classification, rigidity, and quantitative aspects"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.04635","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":"sha256:4769f70daf93911276355c95070bab7350a7bb406e903efcaef2377762e74711","target":"record","created_at":"2026-07-05T07:31:47Z","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":"380323c8d9cc027e49e839a41d86b2baab27bd7257409118203e34318dd627a6","cross_cats_sorted":["math.DS","math.GT","math.OA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2024-01-09T16:05:32Z","title_canon_sha256":"e0ca828ae060e3f9de7098ad2386adfd0c98e317b8d01f58ee3e8a3d90c187bd"},"schema_version":"1.0","source":{"id":"2401.04635","kind":"arxiv","version":1}},"canonical_sha256":"d5e425abcc6bc6a8be12a2f7deb3eb75043f5cb3dd595d2d3870c7b252033be5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d5e425abcc6bc6a8be12a2f7deb3eb75043f5cb3dd595d2d3870c7b252033be5","first_computed_at":"2026-07-05T07:31:47.445395Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:31:47.445395Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"puhNx/l1MyN/pEVJMhOMZJj+OuBR15qMzEOWpOxLeeSxwJXnSkJ9lA/+3JP7EKat1iCgu4h5CKx4Tr6tFbxACQ==","signature_status":"signed_v1","signed_at":"2026-07-05T07:31:47.445748Z","signed_message":"canonical_sha256_bytes"},"source_id":"2401.04635","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4769f70daf93911276355c95070bab7350a7bb406e903efcaef2377762e74711","sha256:ef0609407ad037ffc6491788271ba9604b799b92492f10ccb60a353dd59b13e1"],"state_sha256":"cc37051740e5c13ffe5e2220645b8d46d67f162cd2cfb5779e432d3d2b6bef1f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DyebvS/eectFuGMBfEdhS+H4vKP2h7+afIwmfZjl+IRC7X/SVnnu4FlGevJy846C0DeYCpqB9vMaEWWABAxfBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T16:54:04.646418Z","bundle_sha256":"001af881f53dacb6383f59888e1f6146cfa7822b5872f0e2ec7f0a35099d3691"}}