{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:PSGJOXA4TSW2VRFNYCC4STR6JG","short_pith_number":"pith:PSGJOXA4","canonical_record":{"source":{"id":"1604.05885","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-04-20T10:15:01Z","cross_cats_sorted":[],"title_canon_sha256":"ee097b66ed7e922a77ae79806ce8f97d5bdde926b6533936cafbed6037eafb62","abstract_canon_sha256":"be6fb3c6ac7792bc1035e028663db3bf43ee6d04dd8ef75a20b05b5d405b3493"},"schema_version":"1.0"},"canonical_sha256":"7c8c975c1c9cadaac4adc085c94e3e49b02a4fa385e497f68d3138f0cf8d2c4b","source":{"kind":"arxiv","id":"1604.05885","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.05885","created_at":"2026-05-18T01:16:34Z"},{"alias_kind":"arxiv_version","alias_value":"1604.05885v1","created_at":"2026-05-18T01:16:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.05885","created_at":"2026-05-18T01:16:34Z"},{"alias_kind":"pith_short_12","alias_value":"PSGJOXA4TSW2","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PSGJOXA4TSW2VRFN","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PSGJOXA4","created_at":"2026-05-18T12:30:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:PSGJOXA4TSW2VRFNYCC4STR6JG","target":"record","payload":{"canonical_record":{"source":{"id":"1604.05885","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-04-20T10:15:01Z","cross_cats_sorted":[],"title_canon_sha256":"ee097b66ed7e922a77ae79806ce8f97d5bdde926b6533936cafbed6037eafb62","abstract_canon_sha256":"be6fb3c6ac7792bc1035e028663db3bf43ee6d04dd8ef75a20b05b5d405b3493"},"schema_version":"1.0"},"canonical_sha256":"7c8c975c1c9cadaac4adc085c94e3e49b02a4fa385e497f68d3138f0cf8d2c4b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:34.348958Z","signature_b64":"UwfwxAwm312wnp+seCww9az0JT5367r1pLd2GaVz980T4UQ70CBL38EgKMZxfpB7vgxrt1fBgsgJk1822nVRDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c8c975c1c9cadaac4adc085c94e3e49b02a4fa385e497f68d3138f0cf8d2c4b","last_reissued_at":"2026-05-18T01:16:34.348292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:34.348292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.05885","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":"2026-05-18T01:16:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5ZirQjiD0feTampQ6/58cH0B0ws21kWq4u8F8wvTAQGx4KMmCNsIbXPVdHnYMdnE+AZeUO2sxIP6Q7fw1SvuBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T23:42:11.867072Z"},"content_sha256":"e6693f240c7302150127b1d6185b5ecb2155f4418d0f3ba348250d9ebbce9afe","schema_version":"1.0","event_id":"sha256:e6693f240c7302150127b1d6185b5ecb2155f4418d0f3ba348250d9ebbce9afe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:PSGJOXA4TSW2VRFNYCC4STR6JG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Locally compact groups approximable by subgroups isomorphic to $\\mathbb Z$ or $\\mathbb R$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Hatem Hamrouni, Karl H. Hofmann","submitted_at":"2016-04-20T10:15:01Z","abstract_excerpt":"Let $G$ be a locally compact topological group, $G_0$ the connected component of its identity element, and comp(G) the union of all compact subgroups. A topological group will be called inductively monothetic if any subgroup generated (as a topological group) by finitely many elements is generated (as a topological group) by a single element. The space SUB(G) of all closed subgroups of $G$ carries a compact Hausdorff topology called the Chabauty topology. Let $F_1(G)$, respectively, $R_1(G)$, denote the subspace of all discrete subgroups isomorphic to $\\mathbb Z$, respectively, all subgroups i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05885","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":"2026-05-18T01:16:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hQVdFn+Sbx+Jx5WYXE6fEW/pqg2rWhARb17WuQbRGp2lAiLfdDmEdLeSMMr3rB62Ouj9vq32dqYNiE+0XW9XDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T23:42:11.867427Z"},"content_sha256":"544d5006d470f63a1874ec3b5fcfea2fa861e074163d7f54c7ce557474ec9550","schema_version":"1.0","event_id":"sha256:544d5006d470f63a1874ec3b5fcfea2fa861e074163d7f54c7ce557474ec9550"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PSGJOXA4TSW2VRFNYCC4STR6JG/bundle.json","state_url":"https://pith.science/pith/PSGJOXA4TSW2VRFNYCC4STR6JG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PSGJOXA4TSW2VRFNYCC4STR6JG/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":"2026-07-04T23:42:11Z","links":{"resolver":"https://pith.science/pith/PSGJOXA4TSW2VRFNYCC4STR6JG","bundle":"https://pith.science/pith/PSGJOXA4TSW2VRFNYCC4STR6JG/bundle.json","state":"https://pith.science/pith/PSGJOXA4TSW2VRFNYCC4STR6JG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PSGJOXA4TSW2VRFNYCC4STR6JG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:PSGJOXA4TSW2VRFNYCC4STR6JG","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":"be6fb3c6ac7792bc1035e028663db3bf43ee6d04dd8ef75a20b05b5d405b3493","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-04-20T10:15:01Z","title_canon_sha256":"ee097b66ed7e922a77ae79806ce8f97d5bdde926b6533936cafbed6037eafb62"},"schema_version":"1.0","source":{"id":"1604.05885","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.05885","created_at":"2026-05-18T01:16:34Z"},{"alias_kind":"arxiv_version","alias_value":"1604.05885v1","created_at":"2026-05-18T01:16:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.05885","created_at":"2026-05-18T01:16:34Z"},{"alias_kind":"pith_short_12","alias_value":"PSGJOXA4TSW2","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PSGJOXA4TSW2VRFN","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PSGJOXA4","created_at":"2026-05-18T12:30:39Z"}],"graph_snapshots":[{"event_id":"sha256:544d5006d470f63a1874ec3b5fcfea2fa861e074163d7f54c7ce557474ec9550","target":"graph","created_at":"2026-05-18T01:16:34Z","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":"Let $G$ be a locally compact topological group, $G_0$ the connected component of its identity element, and comp(G) the union of all compact subgroups. A topological group will be called inductively monothetic if any subgroup generated (as a topological group) by finitely many elements is generated (as a topological group) by a single element. The space SUB(G) of all closed subgroups of $G$ carries a compact Hausdorff topology called the Chabauty topology. Let $F_1(G)$, respectively, $R_1(G)$, denote the subspace of all discrete subgroups isomorphic to $\\mathbb Z$, respectively, all subgroups i","authors_text":"Hatem Hamrouni, Karl H. Hofmann","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-04-20T10:15:01Z","title":"Locally compact groups approximable by subgroups isomorphic to $\\mathbb Z$ or $\\mathbb R$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05885","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:e6693f240c7302150127b1d6185b5ecb2155f4418d0f3ba348250d9ebbce9afe","target":"record","created_at":"2026-05-18T01:16:34Z","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":"be6fb3c6ac7792bc1035e028663db3bf43ee6d04dd8ef75a20b05b5d405b3493","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-04-20T10:15:01Z","title_canon_sha256":"ee097b66ed7e922a77ae79806ce8f97d5bdde926b6533936cafbed6037eafb62"},"schema_version":"1.0","source":{"id":"1604.05885","kind":"arxiv","version":1}},"canonical_sha256":"7c8c975c1c9cadaac4adc085c94e3e49b02a4fa385e497f68d3138f0cf8d2c4b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7c8c975c1c9cadaac4adc085c94e3e49b02a4fa385e497f68d3138f0cf8d2c4b","first_computed_at":"2026-05-18T01:16:34.348292Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:16:34.348292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UwfwxAwm312wnp+seCww9az0JT5367r1pLd2GaVz980T4UQ70CBL38EgKMZxfpB7vgxrt1fBgsgJk1822nVRDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:16:34.348958Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.05885","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e6693f240c7302150127b1d6185b5ecb2155f4418d0f3ba348250d9ebbce9afe","sha256:544d5006d470f63a1874ec3b5fcfea2fa861e074163d7f54c7ce557474ec9550"],"state_sha256":"5f0cbfceec513b11e7b4fa764e6ae177ca30cdba9a5230b50b199273beef4514"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"50LrJqug627T90mF9D3pU5J77jhcT/yxufpz8wEkIwuc+iSs83MIf3OtQhrzezQ/l5kUdiLqSJAXfGjj6H6cAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T23:42:11.869379Z","bundle_sha256":"b5b340eac237fef687439f02b70e552522dd4f82780188f24ed238f11c485cdc"}}