{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:HZP2YFEJLXBSMZCYOZA6LM4QTV","short_pith_number":"pith:HZP2YFEJ","canonical_record":{"source":{"id":"1210.0336","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-01T10:10:08Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"ba9a127061c4dc09ee44148d881d79591fd0d591fca091fcc4727d43fcb36211","abstract_canon_sha256":"784821c6441ff83c5b08101afb28bfc4b27d8b26a87314eaa7220273a1f2c615"},"schema_version":"1.0"},"canonical_sha256":"3e5fac14895dc32664587641e5b3909d6bf9297cfcb0a356189cb0218f22ecc1","source":{"kind":"arxiv","id":"1210.0336","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.0336","created_at":"2026-05-18T00:15:59Z"},{"alias_kind":"arxiv_version","alias_value":"1210.0336v1","created_at":"2026-05-18T00:15:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.0336","created_at":"2026-05-18T00:15:59Z"},{"alias_kind":"pith_short_12","alias_value":"HZP2YFEJLXBS","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HZP2YFEJLXBSMZCY","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HZP2YFEJ","created_at":"2026-05-18T12:27:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:HZP2YFEJLXBSMZCYOZA6LM4QTV","target":"record","payload":{"canonical_record":{"source":{"id":"1210.0336","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-01T10:10:08Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"ba9a127061c4dc09ee44148d881d79591fd0d591fca091fcc4727d43fcb36211","abstract_canon_sha256":"784821c6441ff83c5b08101afb28bfc4b27d8b26a87314eaa7220273a1f2c615"},"schema_version":"1.0"},"canonical_sha256":"3e5fac14895dc32664587641e5b3909d6bf9297cfcb0a356189cb0218f22ecc1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:59.801015Z","signature_b64":"lvFxY9OhcwPRFIapVYCFs1fRwCLkHl4pyhUydPusw5FqXUGKA6O4pcAmvC7ebivrfobBBi6lhIVscRw9yky+BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e5fac14895dc32664587641e5b3909d6bf9297cfcb0a356189cb0218f22ecc1","last_reissued_at":"2026-05-18T00:15:59.800486Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:59.800486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.0336","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":"2026-05-18T00:15:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7EN0yKdMKCrjwPBE1rRINCiAjLeFgv+otfECHo3cu74iSJm+dghLYSpccfNwCzbrb+lBfkbMvS3AtLjDB18nBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T09:07:45.900951Z"},"content_sha256":"6a0ca1d67efba8df718542f6617be298f2531a6db1c9b16ffa8771285ca0466f","schema_version":"1.0","event_id":"sha256:6a0ca1d67efba8df718542f6617be298f2531a6db1c9b16ffa8771285ca0466f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:HZP2YFEJLXBSMZCYOZA6LM4QTV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"W*-superrigidity for group von Neumann algebras of left-right wreath products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.OA","authors_text":"Mihaita Berbec, Stefaan Vaes","submitted_at":"2012-10-01T10:10:08Z","abstract_excerpt":"We prove that for many nonamenable groups \\Gamma, including all hyperbolic groups and all nontrivial free products, the left-right wreath product group G := (Z/2Z)^(\\Gamma) \\rtimes (\\Gamma \\times \\Gamma) is W*-superrigid. This means that the group von Neumann algebra LG entirely remembers G. More precisely, if LG is isomorphic with L\\Lambda for an arbitrary countable group \\Lambda, then \\Lambda must be isomorphic with G."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0336","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":"2026-05-18T00:15:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xEOPp76WWn2PPL4ApZZ5eRws/6mAyL0pDz7iYD5IJoIWhJr5negocrwA3OKU1kICQ5p1uS8crcyNFWew3yqpDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T09:07:45.901299Z"},"content_sha256":"6f224e3a0721818c62c1a221fbf5cec3481efba22696da0e97f249889227516a","schema_version":"1.0","event_id":"sha256:6f224e3a0721818c62c1a221fbf5cec3481efba22696da0e97f249889227516a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HZP2YFEJLXBSMZCYOZA6LM4QTV/bundle.json","state_url":"https://pith.science/pith/HZP2YFEJLXBSMZCYOZA6LM4QTV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HZP2YFEJLXBSMZCYOZA6LM4QTV/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":"2026-05-28T09:07:45Z","links":{"resolver":"https://pith.science/pith/HZP2YFEJLXBSMZCYOZA6LM4QTV","bundle":"https://pith.science/pith/HZP2YFEJLXBSMZCYOZA6LM4QTV/bundle.json","state":"https://pith.science/pith/HZP2YFEJLXBSMZCYOZA6LM4QTV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HZP2YFEJLXBSMZCYOZA6LM4QTV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:HZP2YFEJLXBSMZCYOZA6LM4QTV","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":"784821c6441ff83c5b08101afb28bfc4b27d8b26a87314eaa7220273a1f2c615","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-01T10:10:08Z","title_canon_sha256":"ba9a127061c4dc09ee44148d881d79591fd0d591fca091fcc4727d43fcb36211"},"schema_version":"1.0","source":{"id":"1210.0336","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.0336","created_at":"2026-05-18T00:15:59Z"},{"alias_kind":"arxiv_version","alias_value":"1210.0336v1","created_at":"2026-05-18T00:15:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.0336","created_at":"2026-05-18T00:15:59Z"},{"alias_kind":"pith_short_12","alias_value":"HZP2YFEJLXBS","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HZP2YFEJLXBSMZCY","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HZP2YFEJ","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:6f224e3a0721818c62c1a221fbf5cec3481efba22696da0e97f249889227516a","target":"graph","created_at":"2026-05-18T00:15:59Z","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 that for many nonamenable groups \\Gamma, including all hyperbolic groups and all nontrivial free products, the left-right wreath product group G := (Z/2Z)^(\\Gamma) \\rtimes (\\Gamma \\times \\Gamma) is W*-superrigid. This means that the group von Neumann algebra LG entirely remembers G. More precisely, if LG is isomorphic with L\\Lambda for an arbitrary countable group \\Lambda, then \\Lambda must be isomorphic with G.","authors_text":"Mihaita Berbec, Stefaan Vaes","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-01T10:10:08Z","title":"W*-superrigidity for group von Neumann algebras of left-right wreath products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0336","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":"sha256:6a0ca1d67efba8df718542f6617be298f2531a6db1c9b16ffa8771285ca0466f","target":"record","created_at":"2026-05-18T00:15:59Z","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":"784821c6441ff83c5b08101afb28bfc4b27d8b26a87314eaa7220273a1f2c615","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-01T10:10:08Z","title_canon_sha256":"ba9a127061c4dc09ee44148d881d79591fd0d591fca091fcc4727d43fcb36211"},"schema_version":"1.0","source":{"id":"1210.0336","kind":"arxiv","version":1}},"canonical_sha256":"3e5fac14895dc32664587641e5b3909d6bf9297cfcb0a356189cb0218f22ecc1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3e5fac14895dc32664587641e5b3909d6bf9297cfcb0a356189cb0218f22ecc1","first_computed_at":"2026-05-18T00:15:59.800486Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:15:59.800486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lvFxY9OhcwPRFIapVYCFs1fRwCLkHl4pyhUydPusw5FqXUGKA6O4pcAmvC7ebivrfobBBi6lhIVscRw9yky+BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:15:59.801015Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.0336","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6a0ca1d67efba8df718542f6617be298f2531a6db1c9b16ffa8771285ca0466f","sha256:6f224e3a0721818c62c1a221fbf5cec3481efba22696da0e97f249889227516a"],"state_sha256":"02fdd2c5cc1c220d8f90469e403f83221bf1655b645cbf56e0da14711f1f6822"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o0dqRoh3Vv0lS6EQkn+IgA4vd1SeHNcIR5vwG68thc/V7MU9geOncVB5IOyej9EfvqbmpsxUwnCvpuiV55eRCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T09:07:45.903409Z","bundle_sha256":"a201e1153974f4dd39223a10fc632afac164f899b92fe1b56f53ca19ead4802d"}}