{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:4KHLJHGTDEG52CG2J6T3VCI6IK","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":"f5f79e78b22175b85994fe414cae07faabc0afc8ab15e4d3158c8fa5c318c6b5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2007-01-31T14:50:04Z","title_canon_sha256":"9d3d554b1160b7aa98404501cbcd67fea19ffceaa42e80f105dc373c66526af6"},"schema_version":"1.0","source":{"id":"math/0701925","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0701925","created_at":"2026-05-18T04:28:37Z"},{"alias_kind":"arxiv_version","alias_value":"math/0701925v3","created_at":"2026-05-18T04:28:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0701925","created_at":"2026-05-18T04:28:37Z"},{"alias_kind":"pith_short_12","alias_value":"4KHLJHGTDEG5","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"4KHLJHGTDEG52CG2","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"4KHLJHGT","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:7819e1c6c29eec7d572ff63667843ad43d71df03ef17796448a9a212a05179b1","target":"graph","created_at":"2026-05-18T04:28:37Z","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":"This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable groups, groups possessing an infinite soluble normal subgroup and virtually free groups. As a tool for the amenable case we generalize Lackenby's trichotomy theorem on finitely presented groups.","authors_text":"Andrei Jaikin-Zapirain, Miklos Abert, Nikolay Nikolov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2007-01-31T14:50:04Z","title":"The rank gradient from a combinatorial viewpoint"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701925","kind":"arxiv","version":3},"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:b987864cc07f655ae0dd72a30c700ba73f52da80f75cb140484044c15110ed11","target":"record","created_at":"2026-05-18T04:28:37Z","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":"f5f79e78b22175b85994fe414cae07faabc0afc8ab15e4d3158c8fa5c318c6b5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2007-01-31T14:50:04Z","title_canon_sha256":"9d3d554b1160b7aa98404501cbcd67fea19ffceaa42e80f105dc373c66526af6"},"schema_version":"1.0","source":{"id":"math/0701925","kind":"arxiv","version":3}},"canonical_sha256":"e28eb49cd3190ddd08da4fa7ba891e42816bba94ba2c42ebe391aa1d64b137ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e28eb49cd3190ddd08da4fa7ba891e42816bba94ba2c42ebe391aa1d64b137ce","first_computed_at":"2026-05-18T04:28:37.151641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:37.151641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qu9teJvC3HjE/liuC7YiMm62C1j/K+D/sLpKMaBCYRDRvTD+7arYqKZxS3928rrjCUvwE1GTSV85tWjuOm3ACg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:37.152109Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0701925","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b987864cc07f655ae0dd72a30c700ba73f52da80f75cb140484044c15110ed11","sha256:7819e1c6c29eec7d572ff63667843ad43d71df03ef17796448a9a212a05179b1"],"state_sha256":"28de5ab729c0e562db58c1a2787aa51f5c3c3f5dc8f142f6ebe54138afb26054"}