{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:TQFVTCARAXTJE2TN5PEBCACBH2","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":"dd19f96e6dfb43ebc047caf5316a8f0d69afb4b78f80421e6667839949bd65ee","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-02-15T22:38:20Z","title_canon_sha256":"9cd860843cb991a5194d516aa22431ed7151eca5b798509e939393bfd4bd614a"},"schema_version":"1.0","source":{"id":"1302.3908","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.3908","created_at":"2026-05-18T02:41:42Z"},{"alias_kind":"arxiv_version","alias_value":"1302.3908v2","created_at":"2026-05-18T02:41:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.3908","created_at":"2026-05-18T02:41:42Z"},{"alias_kind":"pith_short_12","alias_value":"TQFVTCARAXTJ","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"TQFVTCARAXTJE2TN","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"TQFVTCAR","created_at":"2026-05-18T12:28:02Z"}],"graph_snapshots":[{"event_id":"sha256:5682cba681bba1f92eba1a2b9f1b3a32800d74138c5a1bd6a6eab3e7787fff5e","target":"graph","created_at":"2026-05-18T02:41:42Z","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":"Motivated by the usefulness of boundaries in the study of hyperbolic and CAT(0) groups, Bestvina introduced a general approach to group boundaries via the notion of a Z-structure on a group G. Several variations on Z-structures have been studied and existence results have been obtained for some very specific classes of groups. However, little is known about the general question of which groups admit any of the various Z-structures, aside from the (easy) fact that any such G must have type F, i.e., G must admit a finite K(G,1). In fact, Bestvina has asked whether every type F group admits a Z-s","authors_text":"Craig R. Guilbault","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-02-15T22:38:20Z","title":"Weak Z-structures for some classes of groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.3908","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:c3a332880f53e343bb6d8d5d2161a56b6ad8241bfd96b7d9176e6b404c597d50","target":"record","created_at":"2026-05-18T02:41:42Z","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":"dd19f96e6dfb43ebc047caf5316a8f0d69afb4b78f80421e6667839949bd65ee","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-02-15T22:38:20Z","title_canon_sha256":"9cd860843cb991a5194d516aa22431ed7151eca5b798509e939393bfd4bd614a"},"schema_version":"1.0","source":{"id":"1302.3908","kind":"arxiv","version":2}},"canonical_sha256":"9c0b59881105e6926a6debc81100413e9a041dc7bab1c4d8555814f7fae865f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9c0b59881105e6926a6debc81100413e9a041dc7bab1c4d8555814f7fae865f4","first_computed_at":"2026-05-18T02:41:42.565365Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:42.565365Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k2JB9H2xS8MrSVH0A40+Enko5jqaiHfQHemLGnuIJ85L9tZK+nj2E0dDOdRl+42kTAn87sfNH9xZjo6ntL15Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:42.566007Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.3908","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3a332880f53e343bb6d8d5d2161a56b6ad8241bfd96b7d9176e6b404c597d50","sha256:5682cba681bba1f92eba1a2b9f1b3a32800d74138c5a1bd6a6eab3e7787fff5e"],"state_sha256":"0b44469f70dc96c682d63dea09355be0de563933cdbc074b8a916285c5ca1ba0"}