{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:PF32LQO42Z4CWOECYNUDFPMYU5","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":"a8db3fb03979502f545a0c5cf8e9fbf79e164e7d2625263e943504d3f59772eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-08-26T00:17:27Z","title_canon_sha256":"bf5ac8058e733f499c46fd5d26aec02d135c4efae60dd166879df7c362784e22"},"schema_version":"1.0","source":{"id":"1508.06335","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.06335","created_at":"2026-05-18T01:34:43Z"},{"alias_kind":"arxiv_version","alias_value":"1508.06335v1","created_at":"2026-05-18T01:34:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.06335","created_at":"2026-05-18T01:34:43Z"},{"alias_kind":"pith_short_12","alias_value":"PF32LQO42Z4C","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PF32LQO42Z4CWOEC","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PF32LQO4","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:a04b89afeee910543a8b7c1a2d7e3d0d1ef0a10ea6393d7f2cea336ddf38fc89","target":"graph","created_at":"2026-05-18T01:34:43Z","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 initiate the study of the $p$-local commensurability graph of a group, where $p$ is a prime. This graph has vertices consisting of all finite-index subgroups of a group, where an edge is drawn between $A$ and $B$ if $[A : A\\cap B]$ and $[B: A\\cap B]$ are both powers of $p$. We show that any component of the $p$-local commensurability graph of a group with all nilpotent finite quotients is complete. Further, this topological criterion characterizes such groups. In contrast to this result, we show that for any prime $p$ the $p$-local commensurability graph of any large group (e.g. a nonabelia","authors_text":"Daniel Studenmund, Khalid Bou-Rabee","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-08-26T00:17:27Z","title":"The topology of local commensurability graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06335","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:bd1077d7fe8f87fa5cebc46f703cf7ccd95049b1e7e5383fab9f272e209809be","target":"record","created_at":"2026-05-18T01:34:43Z","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":"a8db3fb03979502f545a0c5cf8e9fbf79e164e7d2625263e943504d3f59772eb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-08-26T00:17:27Z","title_canon_sha256":"bf5ac8058e733f499c46fd5d26aec02d135c4efae60dd166879df7c362784e22"},"schema_version":"1.0","source":{"id":"1508.06335","kind":"arxiv","version":1}},"canonical_sha256":"7977a5c1dcd6782b3882c36832bd98a761421993932ff48b358a0a3e88eeea7b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7977a5c1dcd6782b3882c36832bd98a761421993932ff48b358a0a3e88eeea7b","first_computed_at":"2026-05-18T01:34:43.278809Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:43.278809Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ap8kcTZxkv/YiMIynGia+XmnDjIeaopU1szsUXcS9m5hSwuQPnk7YMmYwEGKWKMvtcTBmYnZzDt4WysLZGIZDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:43.279346Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.06335","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bd1077d7fe8f87fa5cebc46f703cf7ccd95049b1e7e5383fab9f272e209809be","sha256:a04b89afeee910543a8b7c1a2d7e3d0d1ef0a10ea6393d7f2cea336ddf38fc89"],"state_sha256":"0aae6f664a5a368280be69ec3cee51dbe3e25e86cacd023d7f2b3ec5e0891bc7"}