{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:QZWHTDZL7SFXFXBD6HG36XCFF6","short_pith_number":"pith:QZWHTDZL","canonical_record":{"source":{"id":"1103.1063","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-03-05T15:46:10Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"db17f34f1242f6514b4586ab09c343cfa40d3d1ecc24f3ca05238ee66821e404","abstract_canon_sha256":"f0a58630a3a4fb04d119c8627b0adecf54e65526001ded2257ebfd3e2b7fb16d"},"schema_version":"1.0"},"canonical_sha256":"866c798f2bfc8b72dc23f1cdbf5c452fb2aac659dfeac2a7f01b739235447746","source":{"kind":"arxiv","id":"1103.1063","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.1063","created_at":"2026-05-18T04:12:11Z"},{"alias_kind":"arxiv_version","alias_value":"1103.1063v2","created_at":"2026-05-18T04:12:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1063","created_at":"2026-05-18T04:12:11Z"},{"alias_kind":"pith_short_12","alias_value":"QZWHTDZL7SFX","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QZWHTDZL7SFXFXBD","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QZWHTDZL","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:QZWHTDZL7SFXFXBD6HG36XCFF6","target":"record","payload":{"canonical_record":{"source":{"id":"1103.1063","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-03-05T15:46:10Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"db17f34f1242f6514b4586ab09c343cfa40d3d1ecc24f3ca05238ee66821e404","abstract_canon_sha256":"f0a58630a3a4fb04d119c8627b0adecf54e65526001ded2257ebfd3e2b7fb16d"},"schema_version":"1.0"},"canonical_sha256":"866c798f2bfc8b72dc23f1cdbf5c452fb2aac659dfeac2a7f01b739235447746","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:11.893885Z","signature_b64":"Ds8FDp9lPh9qwMF/xQAWQPsYqxyQMVzH3jufR5Sv1bAQvzEQYP1WXDbzLtOJ4xYbwMA74xnCZQpnjG8+dQK8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"866c798f2bfc8b72dc23f1cdbf5c452fb2aac659dfeac2a7f01b739235447746","last_reissued_at":"2026-05-18T04:12:11.893441Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:11.893441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.1063","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":"2026-05-18T04:12:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SRhCPNBFIEqbeaVhGN89isRGtkQJACm3mJKwTQV99Jz6RIvGnrKNw3dtOKbsg0OY4hLPtH41Bni0FjdKnWfeBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:33:27.461140Z"},"content_sha256":"7dc52387a4ffc80dd2ca676602e941f1d88626eb09125012b79f1674ef1dc19b","schema_version":"1.0","event_id":"sha256:7dc52387a4ffc80dd2ca676602e941f1d88626eb09125012b79f1674ef1dc19b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:QZWHTDZL7SFXFXBD6HG36XCFF6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bernoulli actions are weakly contained in any free action","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.DS","authors_text":"Benjamin Weiss, Mikl\\'os Ab\\'ert","submitted_at":"2011-03-05T15:46:10Z","abstract_excerpt":"We show that for any countable group, any free probability measure preserving action of the group weakly contains all Bernoulli actions of the group. It follows that for a finitely generated groups, the cost is maximal on Bernoulli actions and that all free factors of i.i.d.-s the group have the same cost.\n  We also show that if a probability measure preserving action f is ergodic, but not strongly ergodic, then f is weakly equivalent to f\\timesI where I denotes the trivial action on the unit interval. This leads to a relative version of the Glasner-Weiss dichotomy."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1063","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":""},"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":"2026-05-18T04:12:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lDqQ//w3bJxXSIVopnsOlPHYJHyGT0i22GpJSP3q3P6B2PMFiAn0kqqKZhADfTCmfFpjFkGm2sLEjBDwgDiUDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:33:27.461879Z"},"content_sha256":"319ada4cf8f02f06af4d23d9ae2e1c5c666f5ad37ea869629e426b504a373ee8","schema_version":"1.0","event_id":"sha256:319ada4cf8f02f06af4d23d9ae2e1c5c666f5ad37ea869629e426b504a373ee8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QZWHTDZL7SFXFXBD6HG36XCFF6/bundle.json","state_url":"https://pith.science/pith/QZWHTDZL7SFXFXBD6HG36XCFF6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QZWHTDZL7SFXFXBD6HG36XCFF6/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":"2026-06-11T23:33:27Z","links":{"resolver":"https://pith.science/pith/QZWHTDZL7SFXFXBD6HG36XCFF6","bundle":"https://pith.science/pith/QZWHTDZL7SFXFXBD6HG36XCFF6/bundle.json","state":"https://pith.science/pith/QZWHTDZL7SFXFXBD6HG36XCFF6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QZWHTDZL7SFXFXBD6HG36XCFF6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:QZWHTDZL7SFXFXBD6HG36XCFF6","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":"f0a58630a3a4fb04d119c8627b0adecf54e65526001ded2257ebfd3e2b7fb16d","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-03-05T15:46:10Z","title_canon_sha256":"db17f34f1242f6514b4586ab09c343cfa40d3d1ecc24f3ca05238ee66821e404"},"schema_version":"1.0","source":{"id":"1103.1063","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.1063","created_at":"2026-05-18T04:12:11Z"},{"alias_kind":"arxiv_version","alias_value":"1103.1063v2","created_at":"2026-05-18T04:12:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1063","created_at":"2026-05-18T04:12:11Z"},{"alias_kind":"pith_short_12","alias_value":"QZWHTDZL7SFX","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QZWHTDZL7SFXFXBD","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QZWHTDZL","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:319ada4cf8f02f06af4d23d9ae2e1c5c666f5ad37ea869629e426b504a373ee8","target":"graph","created_at":"2026-05-18T04:12:11Z","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 show that for any countable group, any free probability measure preserving action of the group weakly contains all Bernoulli actions of the group. It follows that for a finitely generated groups, the cost is maximal on Bernoulli actions and that all free factors of i.i.d.-s the group have the same cost.\n  We also show that if a probability measure preserving action f is ergodic, but not strongly ergodic, then f is weakly equivalent to f\\timesI where I denotes the trivial action on the unit interval. This leads to a relative version of the Glasner-Weiss dichotomy.","authors_text":"Benjamin Weiss, Mikl\\'os Ab\\'ert","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-03-05T15:46:10Z","title":"Bernoulli actions are weakly contained in any free action"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1063","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":"sha256:7dc52387a4ffc80dd2ca676602e941f1d88626eb09125012b79f1674ef1dc19b","target":"record","created_at":"2026-05-18T04:12:11Z","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":"f0a58630a3a4fb04d119c8627b0adecf54e65526001ded2257ebfd3e2b7fb16d","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-03-05T15:46:10Z","title_canon_sha256":"db17f34f1242f6514b4586ab09c343cfa40d3d1ecc24f3ca05238ee66821e404"},"schema_version":"1.0","source":{"id":"1103.1063","kind":"arxiv","version":2}},"canonical_sha256":"866c798f2bfc8b72dc23f1cdbf5c452fb2aac659dfeac2a7f01b739235447746","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"866c798f2bfc8b72dc23f1cdbf5c452fb2aac659dfeac2a7f01b739235447746","first_computed_at":"2026-05-18T04:12:11.893441Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:11.893441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ds8FDp9lPh9qwMF/xQAWQPsYqxyQMVzH3jufR5Sv1bAQvzEQYP1WXDbzLtOJ4xYbwMA74xnCZQpnjG8+dQK8Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:11.893885Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.1063","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7dc52387a4ffc80dd2ca676602e941f1d88626eb09125012b79f1674ef1dc19b","sha256:319ada4cf8f02f06af4d23d9ae2e1c5c666f5ad37ea869629e426b504a373ee8"],"state_sha256":"cc3c8a7c724d9e7f61b671a5d4f8f8dd5f18337f1ac043f7af4d43e76837c885"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"alYUd1C+UeMRoFIG9kwGCToKzaUPCbVSXqweoWkZHvNgczulBTunQEL4JQ9dWN0bHSVivIxDN7qBQV6CqLerBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T23:33:27.466514Z","bundle_sha256":"2f832a5c052d32870b24e24425b9a199dafc01a3c9af1996f12322db0a610020"}}