{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:HLZKHNPVFW3Z4FWFBHI7D2E3VI","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":"0046572d169e5e9d4cbec281c33a0ed4d0f9f84e807d0c9e660a146c2b4a122c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-10-16T17:58:00Z","title_canon_sha256":"9e4186931720b94742cd9fb7095cad856b138bd05bf7a084be5a6379d6478da1"},"schema_version":"1.0","source":{"id":"1710.05904","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.05904","created_at":"2026-05-18T00:32:47Z"},{"alias_kind":"arxiv_version","alias_value":"1710.05904v1","created_at":"2026-05-18T00:32:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.05904","created_at":"2026-05-18T00:32:47Z"},{"alias_kind":"pith_short_12","alias_value":"HLZKHNPVFW3Z","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"HLZKHNPVFW3Z4FWF","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"HLZKHNPV","created_at":"2026-05-18T12:31:18Z"}],"graph_snapshots":[{"event_id":"sha256:eca1bbd93bb1e3994f2fb669ae6ae401ec6564b09050c2a5052553e5637e1fa9","target":"graph","created_at":"2026-05-18T00:32:47Z","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":"Direct powers of perfect groups admit more concise presentations than one might naively suppose. If $H_1G=H_2G=0$, then $G^n$ has a presentation with $O(\\log n)$ generators and $O(\\log n)^3$ relators. If, in addition, there is an element $g\\in G$ that has infinite order in every non-trivial quotient of $G$, then $G^n$ has a presentation with $d(G) +1$ generators and $O(\\log n)$ relators. The bounds that we obtain on the deficiency of $G^n$ are not monotone in $n$; this points to potential counterexamples for the Relation Gap Problem.","authors_text":"Martin R Bridson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-10-16T17:58:00Z","title":"Concise presentations of direct products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05904","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:e675503c2cfd1b720237db277d2c7db94b05031d4f718aaf5cc79813252c550a","target":"record","created_at":"2026-05-18T00:32:47Z","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":"0046572d169e5e9d4cbec281c33a0ed4d0f9f84e807d0c9e660a146c2b4a122c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-10-16T17:58:00Z","title_canon_sha256":"9e4186931720b94742cd9fb7095cad856b138bd05bf7a084be5a6379d6478da1"},"schema_version":"1.0","source":{"id":"1710.05904","kind":"arxiv","version":1}},"canonical_sha256":"3af2a3b5f52db79e16c509d1f1e89baa2d95794ebee1ea6f2b32f38e66eb5286","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3af2a3b5f52db79e16c509d1f1e89baa2d95794ebee1ea6f2b32f38e66eb5286","first_computed_at":"2026-05-18T00:32:47.856018Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:47.856018Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o2b2YYMxjFdQR7VZQwbzZqwayk10l4p9XnITOTHstfHZvgnxgAtkItZwZv5RMy820KPgPKNfYCkWip+hRYPiDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:47.856740Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.05904","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e675503c2cfd1b720237db277d2c7db94b05031d4f718aaf5cc79813252c550a","sha256:eca1bbd93bb1e3994f2fb669ae6ae401ec6564b09050c2a5052553e5637e1fa9"],"state_sha256":"559cf43e1337c508983c9a55235b1222c25ab25c2074189b7be2fab85bc90912"}