{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:NDN7VN5R7AFTFCEHH26ZIBUIYL","short_pith_number":"pith:NDN7VN5R","canonical_record":{"source":{"id":"2103.03696","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2021-03-05T14:15:42Z","cross_cats_sorted":["math.GT","math.OA"],"title_canon_sha256":"ab6579621bd48c79d9989a94d060d8ab00109f55cdb68b58a41996b939b58a3b","abstract_canon_sha256":"1be08b93ec264a5177a10b81ce149399ede7c7d4ddd7fd1ad5c665e2603e7277"},"schema_version":"1.0"},"canonical_sha256":"68dbfab7b1f80b3288873ebd940688c2e00c68944de6abf6ebd89d1f68f41b05","source":{"kind":"arxiv","id":"2103.03696","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2103.03696","created_at":"2026-07-05T10:53:06Z"},{"alias_kind":"arxiv_version","alias_value":"2103.03696v3","created_at":"2026-07-05T10:53:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2103.03696","created_at":"2026-07-05T10:53:06Z"},{"alias_kind":"pith_short_12","alias_value":"NDN7VN5R7AFT","created_at":"2026-07-05T10:53:06Z"},{"alias_kind":"pith_short_16","alias_value":"NDN7VN5R7AFTFCEH","created_at":"2026-07-05T10:53:06Z"},{"alias_kind":"pith_short_8","alias_value":"NDN7VN5R","created_at":"2026-07-05T10:53:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:NDN7VN5R7AFTFCEHH26ZIBUIYL","target":"record","payload":{"canonical_record":{"source":{"id":"2103.03696","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2021-03-05T14:15:42Z","cross_cats_sorted":["math.GT","math.OA"],"title_canon_sha256":"ab6579621bd48c79d9989a94d060d8ab00109f55cdb68b58a41996b939b58a3b","abstract_canon_sha256":"1be08b93ec264a5177a10b81ce149399ede7c7d4ddd7fd1ad5c665e2603e7277"},"schema_version":"1.0"},"canonical_sha256":"68dbfab7b1f80b3288873ebd940688c2e00c68944de6abf6ebd89d1f68f41b05","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:53:06.929719Z","signature_b64":"2TGVsXT3CqQ/NKvmLMrNsY7oKrOvMdMpmbF/kUnLN7y25WMivKQZd8dRzjFIcjK8q7mTp84HYorfDZTv8BnZCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68dbfab7b1f80b3288873ebd940688c2e00c68944de6abf6ebd89d1f68f41b05","last_reissued_at":"2026-07-05T10:53:06.929227Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:53:06.929227Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2103.03696","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:53:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hgNSvb/t81fIrfrZmIeSP1f1FiHJJ8A0IcOfPfyT4g8n4AmZRmbVgL5wpdz8jmCJ447aRXvk86pHaTR1gVOtCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T15:37:19.192008Z"},"content_sha256":"04f69bf1656c5af6ca47898a4ae4558605477a900fe3b8faabe1f599d4e13a81","schema_version":"1.0","event_id":"sha256:04f69bf1656c5af6ca47898a4ae4558605477a900fe3b8faabe1f599d4e13a81"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:NDN7VN5R7AFTFCEHH26ZIBUIYL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Measure equivalence rigidity of $\\mathrm{Out}(F_N)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT","math.OA"],"primary_cat":"math.GR","authors_text":"Camille Horbez, Vincent Guirardel","submitted_at":"2021-03-05T14:15:42Z","abstract_excerpt":"We prove that for every $N\\ge 3$, the group $\\mathrm{Out}(F_N)$ of outer automorphisms of a free group of rank $N$ is superrigid from the point of view of measure equivalence: any countable group that is measure equivalent to $\\mathrm{Out}(F_N)$, is in fact virtually isomorphic to $\\mathrm{Out}(F_N)$.\n  We introduce three new constructions of canonical splittings associated to a subgroup of $\\mathrm{Out}(F_N)$ of independent interest. They encode respectively the collection of invariant free splittings, invariant cyclic splittings, and maximal invariant free factor systems. Our proof also reli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2103.03696","kind":"arxiv","version":3},"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/2103.03696/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":"2026-07-05T10:53:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Sg32JZOI1L9TKbl0cTI+miXhavGEFJPmXtnXGGk3ZBQM2NZrbloBd6GghihOgSyO0SM0XUYJybD4/h8cTtNIBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T15:37:19.192632Z"},"content_sha256":"bd47e26cb4870ed9d22d4a628dcaf591c3461976064a308e6d7e53268146410e","schema_version":"1.0","event_id":"sha256:bd47e26cb4870ed9d22d4a628dcaf591c3461976064a308e6d7e53268146410e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NDN7VN5R7AFTFCEHH26ZIBUIYL/bundle.json","state_url":"https://pith.science/pith/NDN7VN5R7AFTFCEHH26ZIBUIYL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NDN7VN5R7AFTFCEHH26ZIBUIYL/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":"2026-08-09T15:37:19Z","links":{"resolver":"https://pith.science/pith/NDN7VN5R7AFTFCEHH26ZIBUIYL","bundle":"https://pith.science/pith/NDN7VN5R7AFTFCEHH26ZIBUIYL/bundle.json","state":"https://pith.science/pith/NDN7VN5R7AFTFCEHH26ZIBUIYL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NDN7VN5R7AFTFCEHH26ZIBUIYL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:NDN7VN5R7AFTFCEHH26ZIBUIYL","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":"1be08b93ec264a5177a10b81ce149399ede7c7d4ddd7fd1ad5c665e2603e7277","cross_cats_sorted":["math.GT","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2021-03-05T14:15:42Z","title_canon_sha256":"ab6579621bd48c79d9989a94d060d8ab00109f55cdb68b58a41996b939b58a3b"},"schema_version":"1.0","source":{"id":"2103.03696","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2103.03696","created_at":"2026-07-05T10:53:06Z"},{"alias_kind":"arxiv_version","alias_value":"2103.03696v3","created_at":"2026-07-05T10:53:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2103.03696","created_at":"2026-07-05T10:53:06Z"},{"alias_kind":"pith_short_12","alias_value":"NDN7VN5R7AFT","created_at":"2026-07-05T10:53:06Z"},{"alias_kind":"pith_short_16","alias_value":"NDN7VN5R7AFTFCEH","created_at":"2026-07-05T10:53:06Z"},{"alias_kind":"pith_short_8","alias_value":"NDN7VN5R","created_at":"2026-07-05T10:53:06Z"}],"graph_snapshots":[{"event_id":"sha256:bd47e26cb4870ed9d22d4a628dcaf591c3461976064a308e6d7e53268146410e","target":"graph","created_at":"2026-07-05T10:53:06Z","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/2103.03696/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that for every $N\\ge 3$, the group $\\mathrm{Out}(F_N)$ of outer automorphisms of a free group of rank $N$ is superrigid from the point of view of measure equivalence: any countable group that is measure equivalent to $\\mathrm{Out}(F_N)$, is in fact virtually isomorphic to $\\mathrm{Out}(F_N)$.\n  We introduce three new constructions of canonical splittings associated to a subgroup of $\\mathrm{Out}(F_N)$ of independent interest. They encode respectively the collection of invariant free splittings, invariant cyclic splittings, and maximal invariant free factor systems. Our proof also reli","authors_text":"Camille Horbez, Vincent Guirardel","cross_cats":["math.GT","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2021-03-05T14:15:42Z","title":"Measure equivalence rigidity of $\\mathrm{Out}(F_N)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2103.03696","kind":"arxiv","version":3},"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":"sha256:04f69bf1656c5af6ca47898a4ae4558605477a900fe3b8faabe1f599d4e13a81","target":"record","created_at":"2026-07-05T10:53:06Z","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":"1be08b93ec264a5177a10b81ce149399ede7c7d4ddd7fd1ad5c665e2603e7277","cross_cats_sorted":["math.GT","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2021-03-05T14:15:42Z","title_canon_sha256":"ab6579621bd48c79d9989a94d060d8ab00109f55cdb68b58a41996b939b58a3b"},"schema_version":"1.0","source":{"id":"2103.03696","kind":"arxiv","version":3}},"canonical_sha256":"68dbfab7b1f80b3288873ebd940688c2e00c68944de6abf6ebd89d1f68f41b05","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68dbfab7b1f80b3288873ebd940688c2e00c68944de6abf6ebd89d1f68f41b05","first_computed_at":"2026-07-05T10:53:06.929227Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:53:06.929227Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2TGVsXT3CqQ/NKvmLMrNsY7oKrOvMdMpmbF/kUnLN7y25WMivKQZd8dRzjFIcjK8q7mTp84HYorfDZTv8BnZCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:53:06.929719Z","signed_message":"canonical_sha256_bytes"},"source_id":"2103.03696","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:04f69bf1656c5af6ca47898a4ae4558605477a900fe3b8faabe1f599d4e13a81","sha256:bd47e26cb4870ed9d22d4a628dcaf591c3461976064a308e6d7e53268146410e"],"state_sha256":"98463c25abbdcd6255df07eabcaa61269d8525b87287b8e07a428a8426f24d37"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n1VmiiBNXEyCIofZ2b9gdxt7wW7ONsKhB0e3CsiDMmpcJR0QkUz5K0uces6zPPWLy4i+mTfnWzuKie2b0EVNAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T15:37:19.198561Z","bundle_sha256":"e541b840a7f5ac7eb23922f5ed5cae55c18faaaa7f3d85fdf94fb57501f02830"}}