{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:YJBEEJNAFH66VQYHWRAQLBDPBF","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":"d02c3fa97044431ed08dc37084c66a9ccd6ede155f0bcd8c91b4b511eadd6a54","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-08-18T16:59:35Z","title_canon_sha256":"0a7d53c107a607ee41edad2ee84991fe58bf1112f5ea6875af4b83d7ff5cf783"},"schema_version":"1.0","source":{"id":"1608.05332","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.05332","created_at":"2026-05-18T01:08:31Z"},{"alias_kind":"arxiv_version","alias_value":"1608.05332v1","created_at":"2026-05-18T01:08:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.05332","created_at":"2026-05-18T01:08:31Z"},{"alias_kind":"pith_short_12","alias_value":"YJBEEJNAFH66","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"YJBEEJNAFH66VQYH","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"YJBEEJNA","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:6dd105feafaa22b85168e449cf8ead5ba34920e62e874c223f5ed4e3fe23c2fc","target":"graph","created_at":"2026-05-18T01:08:31Z","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":"Equationally compact subgroups of countable groups were introduced by Banaschewski. For all known cases the orbit closure of such a subgroup is a countable subset in the space of subgroups and has finite Cantor-Bendixson rank. We show that there exists a finitely generated group $\\Gamma$ such that for any countable ordinal $\\alpha$ we have an equationally compact subgroup $H\\subset \\Gamma$ for which the Cantor-Bendixson rank of the orbit closure of $H$ equals to $\\alpha+2$. Then we give an explicite construction of continuum many equationally compact subgroups of $\\Gamma$ such that the associa","authors_text":"Gabor Elek, Konrad Krolicki","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-08-18T16:59:35Z","title":"Invariant subsets of the space of subgroups, equational compactness and the weak equivalence of actions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.05332","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":"sha256:62110d8200f6e5e5c66c44f63f4f894280c9db3ca43d5156f1809af610e244dc","target":"record","created_at":"2026-05-18T01:08:31Z","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":"d02c3fa97044431ed08dc37084c66a9ccd6ede155f0bcd8c91b4b511eadd6a54","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-08-18T16:59:35Z","title_canon_sha256":"0a7d53c107a607ee41edad2ee84991fe58bf1112f5ea6875af4b83d7ff5cf783"},"schema_version":"1.0","source":{"id":"1608.05332","kind":"arxiv","version":1}},"canonical_sha256":"c2424225a029fdeac307b44105846f0968ffd98356a973915e1eb48311843b88","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c2424225a029fdeac307b44105846f0968ffd98356a973915e1eb48311843b88","first_computed_at":"2026-05-18T01:08:31.357339Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:31.357339Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UuqgbtnQXHQ12p4flhc8iqUL8AM/5tUROwjjRjKo+Hl2OAN19ahYyzbeRuHBALtoaCCbzcCdJyG6cB8xphKSDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:31.357786Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.05332","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:62110d8200f6e5e5c66c44f63f4f894280c9db3ca43d5156f1809af610e244dc","sha256:6dd105feafaa22b85168e449cf8ead5ba34920e62e874c223f5ed4e3fe23c2fc"],"state_sha256":"64192d4e7f6baff5e8d011aa61ff7920c7f610195f25a68a1eff0648f6fca5d2"}