{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:VGDMGFMCFJIPN6ERZZWCMQMKEZ","short_pith_number":"pith:VGDMGFMC","canonical_record":{"source":{"id":"1903.11486","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-03-27T15:24:49Z","cross_cats_sorted":[],"title_canon_sha256":"a1e1459de1fb424b8513a6e6c53a47ec03f7561cdc190ce6bd64be013189c526","abstract_canon_sha256":"f6a1014c531bdc995732208f2060a079e44da49c890967baa68456c5e4decc73"},"schema_version":"1.0"},"canonical_sha256":"a986c315822a50f6f891ce6c26418a26602115ea3ae01ba601fd84314004513b","source":{"kind":"arxiv","id":"1903.11486","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.11486","created_at":"2026-05-17T23:39:39Z"},{"alias_kind":"arxiv_version","alias_value":"1903.11486v2","created_at":"2026-05-17T23:39:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.11486","created_at":"2026-05-17T23:39:39Z"},{"alias_kind":"pith_short_12","alias_value":"VGDMGFMCFJIP","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VGDMGFMCFJIPN6ER","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VGDMGFMC","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:VGDMGFMCFJIPN6ERZZWCMQMKEZ","target":"record","payload":{"canonical_record":{"source":{"id":"1903.11486","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-03-27T15:24:49Z","cross_cats_sorted":[],"title_canon_sha256":"a1e1459de1fb424b8513a6e6c53a47ec03f7561cdc190ce6bd64be013189c526","abstract_canon_sha256":"f6a1014c531bdc995732208f2060a079e44da49c890967baa68456c5e4decc73"},"schema_version":"1.0"},"canonical_sha256":"a986c315822a50f6f891ce6c26418a26602115ea3ae01ba601fd84314004513b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:39.662108Z","signature_b64":"MPu8KGLpsYm2DltNwoSUg7JuXyd30FHNU2JzfcSmEXOpY+2FrPLyh7Tg1fqd6g/8Y5Aig42x2k/2GoblVZ0qDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a986c315822a50f6f891ce6c26418a26602115ea3ae01ba601fd84314004513b","last_reissued_at":"2026-05-17T23:39:39.661515Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:39.661515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1903.11486","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":"2026-05-17T23:39:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"anlCrRunIeGAD0ihzDnPHSDDHOLzPqck3FIodwVi4Poko00DhnMajrkOIxqQyynkp+jQHZcaMb0nru7mMEEJBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T08:19:55.144272Z"},"content_sha256":"71a99ebb17914e0fd48b820629ff7ea455a81e9e21f07ac4d3b3a82fcfd81116","schema_version":"1.0","event_id":"sha256:71a99ebb17914e0fd48b820629ff7ea455a81e9e21f07ac4d3b3a82fcfd81116"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:VGDMGFMCFJIPN6ERZZWCMQMKEZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Polynomially weighted $\\ell^p$-completions and group homology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexander Engel, Clara Loeh","submitted_at":"2019-03-27T15:24:49Z","abstract_excerpt":"We introduce polynomially weighted $\\ell^p$-norms on the bar complex of a finitely generated group. We prove that, for groups of polynomial or exponential growth, the homology of the completed complex does not depend on the value of $p$ in the range $(1,\\infty)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.11486","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":"2026-05-17T23:39:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mbcIkeZ6nQ58FmpE0pCz4nIjvqfgSS7dmg4nwo+cZYrxQOZYpZ3wnXE6bIElg/QeK79cTsbdKqN5FEW19zjXDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T08:19:55.145012Z"},"content_sha256":"bad37fff77776a44ee46996372f8b0ded89457dee8e6cbb5a7facff328985046","schema_version":"1.0","event_id":"sha256:bad37fff77776a44ee46996372f8b0ded89457dee8e6cbb5a7facff328985046"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VGDMGFMCFJIPN6ERZZWCMQMKEZ/bundle.json","state_url":"https://pith.science/pith/VGDMGFMCFJIPN6ERZZWCMQMKEZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VGDMGFMCFJIPN6ERZZWCMQMKEZ/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":"2026-06-06T08:19:55Z","links":{"resolver":"https://pith.science/pith/VGDMGFMCFJIPN6ERZZWCMQMKEZ","bundle":"https://pith.science/pith/VGDMGFMCFJIPN6ERZZWCMQMKEZ/bundle.json","state":"https://pith.science/pith/VGDMGFMCFJIPN6ERZZWCMQMKEZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VGDMGFMCFJIPN6ERZZWCMQMKEZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:VGDMGFMCFJIPN6ERZZWCMQMKEZ","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":"f6a1014c531bdc995732208f2060a079e44da49c890967baa68456c5e4decc73","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-03-27T15:24:49Z","title_canon_sha256":"a1e1459de1fb424b8513a6e6c53a47ec03f7561cdc190ce6bd64be013189c526"},"schema_version":"1.0","source":{"id":"1903.11486","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.11486","created_at":"2026-05-17T23:39:39Z"},{"alias_kind":"arxiv_version","alias_value":"1903.11486v2","created_at":"2026-05-17T23:39:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.11486","created_at":"2026-05-17T23:39:39Z"},{"alias_kind":"pith_short_12","alias_value":"VGDMGFMCFJIP","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"VGDMGFMCFJIPN6ER","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"VGDMGFMC","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:bad37fff77776a44ee46996372f8b0ded89457dee8e6cbb5a7facff328985046","target":"graph","created_at":"2026-05-17T23:39:39Z","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 introduce polynomially weighted $\\ell^p$-norms on the bar complex of a finitely generated group. We prove that, for groups of polynomial or exponential growth, the homology of the completed complex does not depend on the value of $p$ in the range $(1,\\infty)$.","authors_text":"Alexander Engel, Clara Loeh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-03-27T15:24:49Z","title":"Polynomially weighted $\\ell^p$-completions and group homology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.11486","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":"sha256:71a99ebb17914e0fd48b820629ff7ea455a81e9e21f07ac4d3b3a82fcfd81116","target":"record","created_at":"2026-05-17T23:39:39Z","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":"f6a1014c531bdc995732208f2060a079e44da49c890967baa68456c5e4decc73","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-03-27T15:24:49Z","title_canon_sha256":"a1e1459de1fb424b8513a6e6c53a47ec03f7561cdc190ce6bd64be013189c526"},"schema_version":"1.0","source":{"id":"1903.11486","kind":"arxiv","version":2}},"canonical_sha256":"a986c315822a50f6f891ce6c26418a26602115ea3ae01ba601fd84314004513b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a986c315822a50f6f891ce6c26418a26602115ea3ae01ba601fd84314004513b","first_computed_at":"2026-05-17T23:39:39.661515Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:39.661515Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MPu8KGLpsYm2DltNwoSUg7JuXyd30FHNU2JzfcSmEXOpY+2FrPLyh7Tg1fqd6g/8Y5Aig42x2k/2GoblVZ0qDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:39.662108Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.11486","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71a99ebb17914e0fd48b820629ff7ea455a81e9e21f07ac4d3b3a82fcfd81116","sha256:bad37fff77776a44ee46996372f8b0ded89457dee8e6cbb5a7facff328985046"],"state_sha256":"ce716cdb981b38ade0aec7fb5e78d80fc3621bd3042604c8a2ee999a855cf7c9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AgHp1VMjH8mO0VD11F/daikn0Olqp6XBT4qFvM7Y4jXRMi5WIu2FAgO3J5adOqx3Urabr64ZhidAe/2QchMjDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T08:19:55.149084Z","bundle_sha256":"84393a9aeda52b99e374341457f19bd40ed36d443eda0413fe458c153b204f06"}}