{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:3I7EB7TNVSWCFUI6OHTQBPQDRC","short_pith_number":"pith:3I7EB7TN","canonical_record":{"source":{"id":"1705.00439","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-05-01T07:55:12Z","cross_cats_sorted":["math.GR","math.OA"],"title_canon_sha256":"512f5241291d476542ccbb7a08c628c932ba80844dd0f182b806c02e79a99f8c","abstract_canon_sha256":"6c5c550b75950ea5c376f3614f620f56587895c99f88041f05d832c0f3db494b"},"schema_version":"1.0"},"canonical_sha256":"da3e40fe6dacac22d11e71e700be03889392b9b9063d54b4937d738e753909ea","source":{"kind":"arxiv","id":"1705.00439","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.00439","created_at":"2026-05-18T00:17:54Z"},{"alias_kind":"arxiv_version","alias_value":"1705.00439v4","created_at":"2026-05-18T00:17:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.00439","created_at":"2026-05-18T00:17:54Z"},{"alias_kind":"pith_short_12","alias_value":"3I7EB7TNVSWC","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3I7EB7TNVSWCFUI6","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3I7EB7TN","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:3I7EB7TNVSWCFUI6OHTQBPQDRC","target":"record","payload":{"canonical_record":{"source":{"id":"1705.00439","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-05-01T07:55:12Z","cross_cats_sorted":["math.GR","math.OA"],"title_canon_sha256":"512f5241291d476542ccbb7a08c628c932ba80844dd0f182b806c02e79a99f8c","abstract_canon_sha256":"6c5c550b75950ea5c376f3614f620f56587895c99f88041f05d832c0f3db494b"},"schema_version":"1.0"},"canonical_sha256":"da3e40fe6dacac22d11e71e700be03889392b9b9063d54b4937d738e753909ea","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:54.789227Z","signature_b64":"M+DW1Dw3R1+Bvy+1rPwpHMVv2tuJb2Lcq77YLWhzyutMFBY08IZyzwgR4UfyQ8OmXGKmO+ncrLhoiTWiyxqSAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da3e40fe6dacac22d11e71e700be03889392b9b9063d54b4937d738e753909ea","last_reissued_at":"2026-05-18T00:17:54.788608Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:54.788608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.00439","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:17:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d+OZEVEEXu74X537hPqGtRAQ/mi+PSMlJbtGLwWyGQqMprqMCdr9W2n09YICeAO+Zkb+fAFBHsMca5tQb2iIAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:43:46.193510Z"},"content_sha256":"73f19ec50802d5f7cde7dcfdbcd17b168161eba4fc9d0aec770dfa496ec49dbe","schema_version":"1.0","event_id":"sha256:73f19ec50802d5f7cde7dcfdbcd17b168161eba4fc9d0aec770dfa496ec49dbe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:3I7EB7TNVSWCFUI6OHTQBPQDRC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bernoulli actions of type III_1 and L^2-cohomology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.OA"],"primary_cat":"math.DS","authors_text":"Jonas Wahl, Stefaan Vaes","submitted_at":"2017-05-01T07:55:12Z","abstract_excerpt":"We conjecture that a countable group $G$ admits a nonsingular Bernoulli action of type III$_1$ if and only if the first $L^2$-cohomology of $G$ is nonzero. We prove this conjecture for all groups that admit at least one element of infinite order. We also give numerous explicit examples of type III$_1$ Bernoulli actions of the group of integers and the free groups, with different degrees of ergodicity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.00439","kind":"arxiv","version":4},"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":"2026-05-18T00:17:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k1rbBed4VG0hEV189gXQH3SLvuHpHr8FrD4eQRtqddZ+j0GsRjpMKMA3lzgGvoshJXPY8gEc+tQEG8mMOcUMDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:43:46.194175Z"},"content_sha256":"0c330586d8d224c344f93b1953f7f0404983c7cd044750d6b90b80190a03402b","schema_version":"1.0","event_id":"sha256:0c330586d8d224c344f93b1953f7f0404983c7cd044750d6b90b80190a03402b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3I7EB7TNVSWCFUI6OHTQBPQDRC/bundle.json","state_url":"https://pith.science/pith/3I7EB7TNVSWCFUI6OHTQBPQDRC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3I7EB7TNVSWCFUI6OHTQBPQDRC/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":"2026-05-27T00:43:46Z","links":{"resolver":"https://pith.science/pith/3I7EB7TNVSWCFUI6OHTQBPQDRC","bundle":"https://pith.science/pith/3I7EB7TNVSWCFUI6OHTQBPQDRC/bundle.json","state":"https://pith.science/pith/3I7EB7TNVSWCFUI6OHTQBPQDRC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3I7EB7TNVSWCFUI6OHTQBPQDRC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:3I7EB7TNVSWCFUI6OHTQBPQDRC","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":"6c5c550b75950ea5c376f3614f620f56587895c99f88041f05d832c0f3db494b","cross_cats_sorted":["math.GR","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-05-01T07:55:12Z","title_canon_sha256":"512f5241291d476542ccbb7a08c628c932ba80844dd0f182b806c02e79a99f8c"},"schema_version":"1.0","source":{"id":"1705.00439","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.00439","created_at":"2026-05-18T00:17:54Z"},{"alias_kind":"arxiv_version","alias_value":"1705.00439v4","created_at":"2026-05-18T00:17:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.00439","created_at":"2026-05-18T00:17:54Z"},{"alias_kind":"pith_short_12","alias_value":"3I7EB7TNVSWC","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3I7EB7TNVSWCFUI6","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3I7EB7TN","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:0c330586d8d224c344f93b1953f7f0404983c7cd044750d6b90b80190a03402b","target":"graph","created_at":"2026-05-18T00:17:54Z","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 conjecture that a countable group $G$ admits a nonsingular Bernoulli action of type III$_1$ if and only if the first $L^2$-cohomology of $G$ is nonzero. We prove this conjecture for all groups that admit at least one element of infinite order. We also give numerous explicit examples of type III$_1$ Bernoulli actions of the group of integers and the free groups, with different degrees of ergodicity.","authors_text":"Jonas Wahl, Stefaan Vaes","cross_cats":["math.GR","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-05-01T07:55:12Z","title":"Bernoulli actions of type III_1 and L^2-cohomology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.00439","kind":"arxiv","version":4},"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":"sha256:73f19ec50802d5f7cde7dcfdbcd17b168161eba4fc9d0aec770dfa496ec49dbe","target":"record","created_at":"2026-05-18T00:17:54Z","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":"6c5c550b75950ea5c376f3614f620f56587895c99f88041f05d832c0f3db494b","cross_cats_sorted":["math.GR","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-05-01T07:55:12Z","title_canon_sha256":"512f5241291d476542ccbb7a08c628c932ba80844dd0f182b806c02e79a99f8c"},"schema_version":"1.0","source":{"id":"1705.00439","kind":"arxiv","version":4}},"canonical_sha256":"da3e40fe6dacac22d11e71e700be03889392b9b9063d54b4937d738e753909ea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da3e40fe6dacac22d11e71e700be03889392b9b9063d54b4937d738e753909ea","first_computed_at":"2026-05-18T00:17:54.788608Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:54.788608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M+DW1Dw3R1+Bvy+1rPwpHMVv2tuJb2Lcq77YLWhzyutMFBY08IZyzwgR4UfyQ8OmXGKmO+ncrLhoiTWiyxqSAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:54.789227Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.00439","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73f19ec50802d5f7cde7dcfdbcd17b168161eba4fc9d0aec770dfa496ec49dbe","sha256:0c330586d8d224c344f93b1953f7f0404983c7cd044750d6b90b80190a03402b"],"state_sha256":"321de9211e481df7daddeef6e752cffc7cb4a5a4b8472f59a5218c98a14f6333"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+8Zu5R47RkoTeSRyG4iduTgnkVNoYVqnCCrr8A/1+EA07WXVHydZnMYJRmRzOKFE/DesEPOD1lZWQDahhOdiAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T00:43:46.197696Z","bundle_sha256":"ceaec819c8cd6d463f9afabd00e0f7d2af3c4348a754ad1b00ef1344b8e088aa"}}