{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:HDFKMFXUG5N6MBR6XLWPMDVRIR","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":"9c5868777436a58a13d83367ad7527b925ac56744b31d2445cc6526d77a9b609","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-12-18T22:53:11Z","title_canon_sha256":"7a4c837a9c229d710867cd87d2b0d1072f93febdd3f26cce8eb65d4475c9d986"},"schema_version":"1.0","source":{"id":"1012.4127","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.4127","created_at":"2026-05-18T04:27:27Z"},{"alias_kind":"arxiv_version","alias_value":"1012.4127v2","created_at":"2026-05-18T04:27:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.4127","created_at":"2026-05-18T04:27:27Z"},{"alias_kind":"pith_short_12","alias_value":"HDFKMFXUG5N6","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"HDFKMFXUG5N6MBR6","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"HDFKMFXU","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:bb00d2a2cb4920eb021ec478c34107d9dfea2f446c2019fbc7074fc858ed511e","target":"graph","created_at":"2026-05-18T04:27:27Z","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 prove that for any natural n>1, the abstract commensurator group of the Baumslag - Solitar group BS(1,n) is isomorphic to the group of 2 by 2 upper triangular matrices A over rational numbers with A_{11}=1. We also prove that for any finitely generated group G with the unique root property the natural homomorphisms Aut(G)--> Comm(G)--> QI(G) are embeddings.","authors_text":"Oleg Bogopolski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-12-18T22:53:11Z","title":"Abstract commensurators of solvable Baumslag - Solitar groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.4127","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:fb2f36e4d40e2b19961153d3a43cdd8d1be4b00fc46d456de48bc8bcef3428a8","target":"record","created_at":"2026-05-18T04:27:27Z","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":"9c5868777436a58a13d83367ad7527b925ac56744b31d2445cc6526d77a9b609","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-12-18T22:53:11Z","title_canon_sha256":"7a4c837a9c229d710867cd87d2b0d1072f93febdd3f26cce8eb65d4475c9d986"},"schema_version":"1.0","source":{"id":"1012.4127","kind":"arxiv","version":2}},"canonical_sha256":"38caa616f4375be6063ebaecf60eb1447849a5d3587e3c1357210f9e46a6af2f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"38caa616f4375be6063ebaecf60eb1447849a5d3587e3c1357210f9e46a6af2f","first_computed_at":"2026-05-18T04:27:27.899185Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:27.899185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Gpzuj5mPZKekKuGMHjnj+dzZj3z8HzdvP9iphNYU38HePdM2FqhaHl8Iu4GKh+w42SEZlBK4zmRUwtW8qvLwDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:27.899647Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.4127","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb2f36e4d40e2b19961153d3a43cdd8d1be4b00fc46d456de48bc8bcef3428a8","sha256:bb00d2a2cb4920eb021ec478c34107d9dfea2f446c2019fbc7074fc858ed511e"],"state_sha256":"a3292233c05eeb5f18001fa9fa9b79a52ed44ba304b23420efced72d1d327e01"}