{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:N5LOHF5ZTXU3HLARJBYOZWYEQB","short_pith_number":"pith:N5LOHF5Z","canonical_record":{"source":{"id":"1603.08586","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-03-28T23:04:39Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"df2a2c4aa41cc7104e7d1df816d94a0664ffeab09340c25399168e775dbd6ce6","abstract_canon_sha256":"7ad5cc08ae93cbf7e4d4b145ae4ac57ec28b9f630b2136c0c533df3b3985d24d"},"schema_version":"1.0"},"canonical_sha256":"6f56e397b99de9b3ac114870ecdb04804c3ac2e41cb5cc6a0dd38418b08d8c8b","source":{"kind":"arxiv","id":"1603.08586","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.08586","created_at":"2026-05-18T01:12:54Z"},{"alias_kind":"arxiv_version","alias_value":"1603.08586v2","created_at":"2026-05-18T01:12:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.08586","created_at":"2026-05-18T01:12:54Z"},{"alias_kind":"pith_short_12","alias_value":"N5LOHF5ZTXU3","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"N5LOHF5ZTXU3HLAR","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"N5LOHF5Z","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:N5LOHF5ZTXU3HLARJBYOZWYEQB","target":"record","payload":{"canonical_record":{"source":{"id":"1603.08586","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-03-28T23:04:39Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"df2a2c4aa41cc7104e7d1df816d94a0664ffeab09340c25399168e775dbd6ce6","abstract_canon_sha256":"7ad5cc08ae93cbf7e4d4b145ae4ac57ec28b9f630b2136c0c533df3b3985d24d"},"schema_version":"1.0"},"canonical_sha256":"6f56e397b99de9b3ac114870ecdb04804c3ac2e41cb5cc6a0dd38418b08d8c8b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:54.823033Z","signature_b64":"1MqnJWfwUW2jyk3ruJgfmG4RpH9vcuybob6ShHsOcfmmv2NMCtocF5P/7krEIkQTLGTEsAsMzT8ZdSw/zqF4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f56e397b99de9b3ac114870ecdb04804c3ac2e41cb5cc6a0dd38418b08d8c8b","last_reissued_at":"2026-05-18T01:12:54.822639Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:54.822639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.08586","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/DaGHhz7+uZGwg4J0OjVkJrVvQl8Ma7qUqlo4lByjg546mj0W32BIgMov1MuTJpstQva1NtU8sF8PYRdf7TJBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T04:31:56.003863Z"},"content_sha256":"c79ab0cd8af3a2687e69c1268c2777d051aea14b73eeab220e80e05eadb838e5","schema_version":"1.0","event_id":"sha256:c79ab0cd8af3a2687e69c1268c2777d051aea14b73eeab220e80e05eadb838e5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:N5LOHF5ZTXU3HLARJBYOZWYEQB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Commensurability of groups quasi-isometric to RAAG's","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Jingyin Huang","submitted_at":"2016-03-28T23:04:39Z","abstract_excerpt":"Let $G$ be a right-angled Artin group with defining graph $\\Gamma$ and let $H$ be a finitely generated group quasi-isometric to $G(\\Gamma)$. We show if $G$ satisfies (1) its outer automorphism group is finite; (2) $\\Gamma$ does not have induced 4-cycle; (3) $\\Gamma$ is star-rigid; then $H$ is commensurable to $G$. We show condition (2) is sharp in the sense that if $\\Gamma$ contains an induced 4-cycle, then there exists an $H$ quasi-isometric to $G(\\Gamma)$ but not commensurable to $G(\\Gamma)$. Moreover, one can drop condition (1) if $H$ is a uniform lattice acting on the universal cover of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08586","kind":"arxiv","version":2},"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":"2026-05-18T01:12:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8atiFuebaqe4Wgvyt2bVeCaZHtRaFSkDLHyUGqoAZu/lE7o5g3nSd47IX3UnFPVNwD2jbdvF1jrBeW3hK/9dCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T04:31:56.004236Z"},"content_sha256":"02716193498229f20c4e58a0710ca48f9e1cfd7ad2552cba26aa346c99b33f6b","schema_version":"1.0","event_id":"sha256:02716193498229f20c4e58a0710ca48f9e1cfd7ad2552cba26aa346c99b33f6b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N5LOHF5ZTXU3HLARJBYOZWYEQB/bundle.json","state_url":"https://pith.science/pith/N5LOHF5ZTXU3HLARJBYOZWYEQB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N5LOHF5ZTXU3HLARJBYOZWYEQB/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":"2026-07-04T04:31:56Z","links":{"resolver":"https://pith.science/pith/N5LOHF5ZTXU3HLARJBYOZWYEQB","bundle":"https://pith.science/pith/N5LOHF5ZTXU3HLARJBYOZWYEQB/bundle.json","state":"https://pith.science/pith/N5LOHF5ZTXU3HLARJBYOZWYEQB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N5LOHF5ZTXU3HLARJBYOZWYEQB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:N5LOHF5ZTXU3HLARJBYOZWYEQB","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":"7ad5cc08ae93cbf7e4d4b145ae4ac57ec28b9f630b2136c0c533df3b3985d24d","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-03-28T23:04:39Z","title_canon_sha256":"df2a2c4aa41cc7104e7d1df816d94a0664ffeab09340c25399168e775dbd6ce6"},"schema_version":"1.0","source":{"id":"1603.08586","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.08586","created_at":"2026-05-18T01:12:54Z"},{"alias_kind":"arxiv_version","alias_value":"1603.08586v2","created_at":"2026-05-18T01:12:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.08586","created_at":"2026-05-18T01:12:54Z"},{"alias_kind":"pith_short_12","alias_value":"N5LOHF5ZTXU3","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"N5LOHF5ZTXU3HLAR","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"N5LOHF5Z","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:02716193498229f20c4e58a0710ca48f9e1cfd7ad2552cba26aa346c99b33f6b","target":"graph","created_at":"2026-05-18T01:12:54Z","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 right-angled Artin group with defining graph $\\Gamma$ and let $H$ be a finitely generated group quasi-isometric to $G(\\Gamma)$. We show if $G$ satisfies (1) its outer automorphism group is finite; (2) $\\Gamma$ does not have induced 4-cycle; (3) $\\Gamma$ is star-rigid; then $H$ is commensurable to $G$. We show condition (2) is sharp in the sense that if $\\Gamma$ contains an induced 4-cycle, then there exists an $H$ quasi-isometric to $G(\\Gamma)$ but not commensurable to $G(\\Gamma)$. Moreover, one can drop condition (1) if $H$ is a uniform lattice acting on the universal cover of th","authors_text":"Jingyin Huang","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-03-28T23:04:39Z","title":"Commensurability of groups quasi-isometric to RAAG's"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08586","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:c79ab0cd8af3a2687e69c1268c2777d051aea14b73eeab220e80e05eadb838e5","target":"record","created_at":"2026-05-18T01:12:54Z","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":"7ad5cc08ae93cbf7e4d4b145ae4ac57ec28b9f630b2136c0c533df3b3985d24d","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-03-28T23:04:39Z","title_canon_sha256":"df2a2c4aa41cc7104e7d1df816d94a0664ffeab09340c25399168e775dbd6ce6"},"schema_version":"1.0","source":{"id":"1603.08586","kind":"arxiv","version":2}},"canonical_sha256":"6f56e397b99de9b3ac114870ecdb04804c3ac2e41cb5cc6a0dd38418b08d8c8b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6f56e397b99de9b3ac114870ecdb04804c3ac2e41cb5cc6a0dd38418b08d8c8b","first_computed_at":"2026-05-18T01:12:54.822639Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:54.822639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1MqnJWfwUW2jyk3ruJgfmG4RpH9vcuybob6ShHsOcfmmv2NMCtocF5P/7krEIkQTLGTEsAsMzT8ZdSw/zqF4BA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:54.823033Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.08586","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c79ab0cd8af3a2687e69c1268c2777d051aea14b73eeab220e80e05eadb838e5","sha256:02716193498229f20c4e58a0710ca48f9e1cfd7ad2552cba26aa346c99b33f6b"],"state_sha256":"e465ab8de79ff9d3c7c8035af5276ec9980197ec41da46a15348cd46dd2bfb4d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tp9PVieJidxaU3fkK5kpF5ngA7zCVhaSWLAeG6IN8tjVWpY7ZEo5AiewrOnudqfFTY4oX0Tk9nepPb+s06QCCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T04:31:56.006468Z","bundle_sha256":"c31df1d1184cf9c35bcd6df92b8eaeb9a8f7a2aab684a8a013327c06bee803ea"}}