{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:TMPPDP2XDI3EVAC64PAJUVFUYL","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":"127c9ebef16493c298ba249889ecb5ac916ab715bd7c2e3a4bc9af5a1b68bc68","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-05-18T05:58:40Z","title_canon_sha256":"47c69cb66d51d121e990143a39ae5e59dd557d30bcf3c724fcee3e3072bf44e2"},"schema_version":"1.0","source":{"id":"1805.08071","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.08071","created_at":"2026-05-17T23:50:59Z"},{"alias_kind":"arxiv_version","alias_value":"1805.08071v2","created_at":"2026-05-17T23:50:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.08071","created_at":"2026-05-17T23:50:59Z"},{"alias_kind":"pith_short_12","alias_value":"TMPPDP2XDI3E","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_16","alias_value":"TMPPDP2XDI3EVAC6","created_at":"2026-05-18T12:32:53Z"},{"alias_kind":"pith_short_8","alias_value":"TMPPDP2X","created_at":"2026-05-18T12:32:53Z"}],"graph_snapshots":[{"event_id":"sha256:8a45a9d64f89a7116b832a49052daeec1d507ab9842ef6e78c491654f7f66c26","target":"graph","created_at":"2026-05-17T23:50:59Z","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 describe solutions of the equation $x^ny^m=a^nb^m$ in acylindrically hyperbolic groups (AH-groups), where $a,b$ are non-commensurable special loxodromic elements and $n,m$ are integers with sufficiently large common divisor. Using this description and certain test words in AH-groups, we study the verbal closedness of AH-subgroups in groups. A subgroup $H$ of a group $G$ is called verbally closed if for any word $w(x_1,\\dots, x_n)$ in variables $x_1,\\dots,x_n$ and any element $h\\in H$, the equation $w(x_1,\\dots, x_n)=h$ has a solution in $G$ if and only if it has a solution in $H$. Main Theo","authors_text":"Oleg Bogopolski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-05-18T05:58:40Z","title":"Equations in acylindrically hyperbolic groups and verbal closedness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.08071","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":"sha256:7b3fe200a8a5cf2f051572becf88b649c7a63301433d60bb1f116f1bf4879c32","target":"record","created_at":"2026-05-17T23:50:59Z","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":"127c9ebef16493c298ba249889ecb5ac916ab715bd7c2e3a4bc9af5a1b68bc68","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-05-18T05:58:40Z","title_canon_sha256":"47c69cb66d51d121e990143a39ae5e59dd557d30bcf3c724fcee3e3072bf44e2"},"schema_version":"1.0","source":{"id":"1805.08071","kind":"arxiv","version":2}},"canonical_sha256":"9b1ef1bf571a364a805ee3c09a54b4c2e48c96c0619d56391c94b1a20a1c85cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b1ef1bf571a364a805ee3c09a54b4c2e48c96c0619d56391c94b1a20a1c85cd","first_computed_at":"2026-05-17T23:50:59.682587Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:59.682587Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CmGp2Ik2Bz1g/DG+JlMkUytFzcd3JdJeXjLMos576/HbQsBNYyjpnsNPQc8/4FFftZB74GouUGJA1fWsrtmJCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:59.683254Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.08071","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7b3fe200a8a5cf2f051572becf88b649c7a63301433d60bb1f116f1bf4879c32","sha256:8a45a9d64f89a7116b832a49052daeec1d507ab9842ef6e78c491654f7f66c26"],"state_sha256":"1153f56500c6d338750698e521d419091756b2819257501dc01fdc48a795528c"}