{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:JR5DLQVICHF2QRVOR66L6QV754","short_pith_number":"pith:JR5DLQVI","canonical_record":{"source":{"id":"1001.2793","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-15T23:50:06Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"ec3f31a909904d9f82eea331fa28b4212cd0561828e0481e6baae178f1891dca","abstract_canon_sha256":"a918764c9a7b25c01d10a55f18c839abb57ca4c2fa7e99b166065a9670d574c9"},"schema_version":"1.0"},"canonical_sha256":"4c7a35c2a811cba846ae8fbcbf42bfef2a3161f968147c056af16f017fc2ba90","source":{"kind":"arxiv","id":"1001.2793","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.2793","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"arxiv_version","alias_value":"1001.2793v2","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.2793","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"pith_short_12","alias_value":"JR5DLQVICHF2","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"JR5DLQVICHF2QRVO","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"JR5DLQVI","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:JR5DLQVICHF2QRVOR66L6QV754","target":"record","payload":{"canonical_record":{"source":{"id":"1001.2793","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-15T23:50:06Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"ec3f31a909904d9f82eea331fa28b4212cd0561828e0481e6baae178f1891dca","abstract_canon_sha256":"a918764c9a7b25c01d10a55f18c839abb57ca4c2fa7e99b166065a9670d574c9"},"schema_version":"1.0"},"canonical_sha256":"4c7a35c2a811cba846ae8fbcbf42bfef2a3161f968147c056af16f017fc2ba90","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:46.300068Z","signature_b64":"47HbZ6mhdareTykeVHPo0I5+Cfd0rBLMI2/IH1TpGGXhVQJEcW873Evf+gX/ELU/WB0Xb9ICaIUFa7UkEvo7DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c7a35c2a811cba846ae8fbcbf42bfef2a3161f968147c056af16f017fc2ba90","last_reissued_at":"2026-05-18T04:18:46.299547Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:46.299547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1001.2793","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":"2026-05-18T04:18:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CvawOqZC4NtiRbx4Z48AJECFPseU5SlGTUI81T11CNFkEoaAcyi+N4+xR32OzcTMNk+XzR+BbonLDX4hAysLCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T05:04:02.156364Z"},"content_sha256":"938198db1a1c2c352c1cd61ff0f5579082ca726597c2e86c43c72260e7cd4d38","schema_version":"1.0","event_id":"sha256:938198db1a1c2c352c1cd61ff0f5579082ca726597c2e86c43c72260e7cd4d38"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:JR5DLQVICHF2QRVOR66L6QV754","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal F{\\o}lner foliations are amenable","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.DS","authors_text":"Ana Rechtman, Fernando Alcalde Cuesta","submitted_at":"2010-01-15T23:50:06Z","abstract_excerpt":"For finitely generated groups, amenability and F{\\o}lner properties are equivalent. However, contrary to a widespread idea, Kaimanovich showed that F{\\o}lner condition does not imply amenability for discrete measured equivalence relations. In this paper, we exhibit two examples of $C^\\infty$ foliations of closed manifolds that are F{\\o}lner and non amenable with respect to a finite transverse invariant measure and a transverse invariant volume, respectively. We also prove the equivalence between the two notions when the foliation is minimal, that is all the leaves are dense, giving a positive "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.2793","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":"2026-05-18T04:18:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WtZ2R+b2qz+bWIpSfcQCt8i29x2FYKM1O8AluuZ/8HgpwV0LLd/jmwwt4GSUQ2c2Uf0qCprw02/ZpAbotML8Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T05:04:02.156719Z"},"content_sha256":"15584899cfc90de765346265e00f04e46dac4d460c61aa4287edace1ecea3a9a","schema_version":"1.0","event_id":"sha256:15584899cfc90de765346265e00f04e46dac4d460c61aa4287edace1ecea3a9a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JR5DLQVICHF2QRVOR66L6QV754/bundle.json","state_url":"https://pith.science/pith/JR5DLQVICHF2QRVOR66L6QV754/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JR5DLQVICHF2QRVOR66L6QV754/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":"2026-06-02T05:04:02Z","links":{"resolver":"https://pith.science/pith/JR5DLQVICHF2QRVOR66L6QV754","bundle":"https://pith.science/pith/JR5DLQVICHF2QRVOR66L6QV754/bundle.json","state":"https://pith.science/pith/JR5DLQVICHF2QRVOR66L6QV754/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JR5DLQVICHF2QRVOR66L6QV754/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:JR5DLQVICHF2QRVOR66L6QV754","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":"a918764c9a7b25c01d10a55f18c839abb57ca4c2fa7e99b166065a9670d574c9","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-15T23:50:06Z","title_canon_sha256":"ec3f31a909904d9f82eea331fa28b4212cd0561828e0481e6baae178f1891dca"},"schema_version":"1.0","source":{"id":"1001.2793","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.2793","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"arxiv_version","alias_value":"1001.2793v2","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.2793","created_at":"2026-05-18T04:18:46Z"},{"alias_kind":"pith_short_12","alias_value":"JR5DLQVICHF2","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"JR5DLQVICHF2QRVO","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"JR5DLQVI","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:15584899cfc90de765346265e00f04e46dac4d460c61aa4287edace1ecea3a9a","target":"graph","created_at":"2026-05-18T04:18:46Z","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":"For finitely generated groups, amenability and F{\\o}lner properties are equivalent. However, contrary to a widespread idea, Kaimanovich showed that F{\\o}lner condition does not imply amenability for discrete measured equivalence relations. In this paper, we exhibit two examples of $C^\\infty$ foliations of closed manifolds that are F{\\o}lner and non amenable with respect to a finite transverse invariant measure and a transverse invariant volume, respectively. We also prove the equivalence between the two notions when the foliation is minimal, that is all the leaves are dense, giving a positive ","authors_text":"Ana Rechtman, Fernando Alcalde Cuesta","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-15T23:50:06Z","title":"Minimal F{\\o}lner foliations are amenable"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.2793","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":"sha256:938198db1a1c2c352c1cd61ff0f5579082ca726597c2e86c43c72260e7cd4d38","target":"record","created_at":"2026-05-18T04:18:46Z","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":"a918764c9a7b25c01d10a55f18c839abb57ca4c2fa7e99b166065a9670d574c9","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-15T23:50:06Z","title_canon_sha256":"ec3f31a909904d9f82eea331fa28b4212cd0561828e0481e6baae178f1891dca"},"schema_version":"1.0","source":{"id":"1001.2793","kind":"arxiv","version":2}},"canonical_sha256":"4c7a35c2a811cba846ae8fbcbf42bfef2a3161f968147c056af16f017fc2ba90","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4c7a35c2a811cba846ae8fbcbf42bfef2a3161f968147c056af16f017fc2ba90","first_computed_at":"2026-05-18T04:18:46.299547Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:46.299547Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"47HbZ6mhdareTykeVHPo0I5+Cfd0rBLMI2/IH1TpGGXhVQJEcW873Evf+gX/ELU/WB0Xb9ICaIUFa7UkEvo7DA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:46.300068Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.2793","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:938198db1a1c2c352c1cd61ff0f5579082ca726597c2e86c43c72260e7cd4d38","sha256:15584899cfc90de765346265e00f04e46dac4d460c61aa4287edace1ecea3a9a"],"state_sha256":"462ab5141a8cac23a14c5ffeed0ac149398f4139adc8728937812238ee5968db"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hfu/WEBmdpgDJsrqB7tm5Yn6/zJDu3S4BEj7X3XZs/n3j00T6PmLLAbojyTf4I9KIsoZyBrRzp5lQPGwrwHZAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T05:04:02.158737Z","bundle_sha256":"4d0490806d9f91fe6dbfdaa8f8d92943858976c375e8ef3fc966db9b3bf8aaf0"}}