{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:HBMARECRLTJADKUJT7RRKOSUWZ","short_pith_number":"pith:HBMARECR","canonical_record":{"source":{"id":"1503.01736","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-05T19:18:25Z","cross_cats_sorted":[],"title_canon_sha256":"90286895ed89f079edf94c5e662efddbbf0005a3e019abc54ebeb5b6d3b5aa0d","abstract_canon_sha256":"eb519c6053dad9a28aa88be16c1d8168f1cf057efc7e1476124338b54c939c42"},"schema_version":"1.0"},"canonical_sha256":"38580890515cd201aa899fe3153a54b6498c5c0316b605080ce5e1275b6cdf7a","source":{"kind":"arxiv","id":"1503.01736","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01736","created_at":"2026-05-18T02:25:33Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01736v1","created_at":"2026-05-18T02:25:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01736","created_at":"2026-05-18T02:25:33Z"},{"alias_kind":"pith_short_12","alias_value":"HBMARECRLTJA","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"HBMARECRLTJADKUJ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"HBMARECR","created_at":"2026-05-18T12:29:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:HBMARECRLTJADKUJT7RRKOSUWZ","target":"record","payload":{"canonical_record":{"source":{"id":"1503.01736","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-05T19:18:25Z","cross_cats_sorted":[],"title_canon_sha256":"90286895ed89f079edf94c5e662efddbbf0005a3e019abc54ebeb5b6d3b5aa0d","abstract_canon_sha256":"eb519c6053dad9a28aa88be16c1d8168f1cf057efc7e1476124338b54c939c42"},"schema_version":"1.0"},"canonical_sha256":"38580890515cd201aa899fe3153a54b6498c5c0316b605080ce5e1275b6cdf7a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:33.044898Z","signature_b64":"014bWejYjlE5vJ9hpVQHrKh4j/SMyh1JAnbwNakYJCkGlDZVTGtkoCb5adfGUpTdcDOi1f/rJD9dMh17BPM6Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38580890515cd201aa899fe3153a54b6498c5c0316b605080ce5e1275b6cdf7a","last_reissued_at":"2026-05-18T02:25:33.044436Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:33.044436Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.01736","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:25:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lkiiyCkKodOIIDWrE2YQ7AybuMGwuMCJpVApykwTrgqNWvsK+5cPl4k+dh6G4Z620TkAnAscuYbG6Dl/C8IVCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T13:20:18.308814Z"},"content_sha256":"b0db1b63551ac7447e66b75bb00c52f1ac767fb3906689118ad540670085657a","schema_version":"1.0","event_id":"sha256:b0db1b63551ac7447e66b75bb00c52f1ac767fb3906689118ad540670085657a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:HBMARECRLTJADKUJT7RRKOSUWZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Left relatively convex subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Warren Dicks, Yago Antol\\'in, Zoran Sunic","submitted_at":"2015-03-05T19:18:25Z","abstract_excerpt":"Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order; when G is left orderable, this holds if and only if H is convex in G under some left ordering of G.\n  We give a criterion for H to be left relatively convex in G that generalizes a famous theorem of Burns and Hale and has essentially the same proof. We show that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups, and in surface groups that are not the Klein-bottle group. The free-group case extends a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01736","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:25:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jjlirG2IdQYSH0Qo9/uE2cnRaR0zb7Uv5gXnCVN6oF/42/a1KePaVPfmZ+BNfBwBk+S/1RkNzmYe7p9Prvs8AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T13:20:18.309168Z"},"content_sha256":"3b71a768acd76e48d22b08c6120362ba327753751d8e776f44e734914f9756a6","schema_version":"1.0","event_id":"sha256:3b71a768acd76e48d22b08c6120362ba327753751d8e776f44e734914f9756a6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HBMARECRLTJADKUJT7RRKOSUWZ/bundle.json","state_url":"https://pith.science/pith/HBMARECRLTJADKUJT7RRKOSUWZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HBMARECRLTJADKUJT7RRKOSUWZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-21T13:20:18Z","links":{"resolver":"https://pith.science/pith/HBMARECRLTJADKUJT7RRKOSUWZ","bundle":"https://pith.science/pith/HBMARECRLTJADKUJT7RRKOSUWZ/bundle.json","state":"https://pith.science/pith/HBMARECRLTJADKUJT7RRKOSUWZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HBMARECRLTJADKUJT7RRKOSUWZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:HBMARECRLTJADKUJT7RRKOSUWZ","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":"eb519c6053dad9a28aa88be16c1d8168f1cf057efc7e1476124338b54c939c42","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-05T19:18:25Z","title_canon_sha256":"90286895ed89f079edf94c5e662efddbbf0005a3e019abc54ebeb5b6d3b5aa0d"},"schema_version":"1.0","source":{"id":"1503.01736","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01736","created_at":"2026-05-18T02:25:33Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01736v1","created_at":"2026-05-18T02:25:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01736","created_at":"2026-05-18T02:25:33Z"},{"alias_kind":"pith_short_12","alias_value":"HBMARECRLTJA","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"HBMARECRLTJADKUJ","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"HBMARECR","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:3b71a768acd76e48d22b08c6120362ba327753751d8e776f44e734914f9756a6","target":"graph","created_at":"2026-05-18T02:25:33Z","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":"Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order; when G is left orderable, this holds if and only if H is convex in G under some left ordering of G.\n  We give a criterion for H to be left relatively convex in G that generalizes a famous theorem of Burns and Hale and has essentially the same proof. We show that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups, and in surface groups that are not the Klein-bottle group. The free-group case extends a ","authors_text":"Warren Dicks, Yago Antol\\'in, Zoran Sunic","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-05T19:18:25Z","title":"Left relatively convex subgroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01736","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:b0db1b63551ac7447e66b75bb00c52f1ac767fb3906689118ad540670085657a","target":"record","created_at":"2026-05-18T02:25:33Z","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":"eb519c6053dad9a28aa88be16c1d8168f1cf057efc7e1476124338b54c939c42","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-05T19:18:25Z","title_canon_sha256":"90286895ed89f079edf94c5e662efddbbf0005a3e019abc54ebeb5b6d3b5aa0d"},"schema_version":"1.0","source":{"id":"1503.01736","kind":"arxiv","version":1}},"canonical_sha256":"38580890515cd201aa899fe3153a54b6498c5c0316b605080ce5e1275b6cdf7a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"38580890515cd201aa899fe3153a54b6498c5c0316b605080ce5e1275b6cdf7a","first_computed_at":"2026-05-18T02:25:33.044436Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:33.044436Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"014bWejYjlE5vJ9hpVQHrKh4j/SMyh1JAnbwNakYJCkGlDZVTGtkoCb5adfGUpTdcDOi1f/rJD9dMh17BPM6Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:33.044898Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.01736","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b0db1b63551ac7447e66b75bb00c52f1ac767fb3906689118ad540670085657a","sha256:3b71a768acd76e48d22b08c6120362ba327753751d8e776f44e734914f9756a6"],"state_sha256":"a753711f2e41bf4aace624bf4cb408de8734221d29c2627fb8c45b827d6bb624"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ghFsX4iXT21rp7nm9atBkZqcNW44GfWg/yCIoh0AsucR49wdXNGbG2wfG/gt+UZg8CO+SDh7JIiasJb/lZSLBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T13:20:18.311129Z","bundle_sha256":"453d44bdaefd2d818c2f965546e1ee7b6112292d8ea87e19cd984d43e78b4252"}}