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Consider the abelian group $\\mathcal{A}_W$ generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $\\mathcal{A}_B$ can be defined similarly, using black triangles. These groups are related in the following manner $\\mathcal{A}_W\\cong\\mathcal{A}_B\\cong\\mathbb{Z}\\oplus\\mathbb{Z}\\oplus\\mathcal{C}$ where $\\mathcal{C}$ is a finite abelian group.\n  The finite torsion subgroup $\\mathcal{C}$ is referre"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1408.2984","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-13T11:59:41Z","cross_cats_sorted":[],"title_canon_sha256":"736574bb98b9df8a6604ce56239fbf2e0d12ccd1f01fb75dded633f2cbe2cd1a","abstract_canon_sha256":"f8cf70d00307e6749f9a068a280c5b0e3f89c29d8c131eca1cfd3cc17ddef59e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:19.060639Z","signature_b64":"P8x5peXRENnxmMgVJURKPSU/zzA31WSXMvw+pCbNhOW0dv/LEqnnzCTyhtjf3j8qzTliyuwcFrOHsZB9jtYbAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9125b86196a9cf3dc800b0f22682444413aa5b47a81af95f655923c48fb145f7","last_reissued_at":"2026-05-18T01:22:19.059934Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:19.059934Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Growth rates of groups associated with face 2-coloured triangulations and directed Eulerian digraphs on the sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Thomas A. McCourt","submitted_at":"2014-08-13T11:59:41Z","abstract_excerpt":"Let $\\mathcal{G}$ be a properly face 2-coloured (say black and white) \\break piecewise-linear triangulation of the sphere with vertex set $V$. Consider the abelian group $\\mathcal{A}_W$ generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $\\mathcal{A}_B$ can be defined similarly, using black triangles. These groups are related in the following manner $\\mathcal{A}_W\\cong\\mathcal{A}_B\\cong\\mathbb{Z}\\oplus\\mathbb{Z}\\oplus\\mathcal{C}$ where $\\mathcal{C}$ is a finite abelian group.\n  The finite torsion subgroup $\\mathcal{C}$ is referre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.2984","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1408.2984","created_at":"2026-05-18T01:22:19.060045+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.2984v4","created_at":"2026-05-18T01:22:19.060045+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.2984","created_at":"2026-05-18T01:22:19.060045+00:00"},{"alias_kind":"pith_short_12","alias_value":"SES3QYMWVHHT","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_16","alias_value":"SES3QYMWVHHT3SAA","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_8","alias_value":"SES3QYMW","created_at":"2026-05-18T12:28:49.207871+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SES3QYMWVHHT3SAAWDZCNASEIQ","json":"https://pith.science/pith/SES3QYMWVHHT3SAAWDZCNASEIQ.json","graph_json":"https://pith.science/api/pith-number/SES3QYMWVHHT3SAAWDZCNASEIQ/graph.json","events_json":"https://pith.science/api/pith-number/SES3QYMWVHHT3SAAWDZCNASEIQ/events.json","paper":"https://pith.science/paper/SES3QYMW"},"agent_actions":{"view_html":"https://pith.science/pith/SES3QYMWVHHT3SAAWDZCNASEIQ","download_json":"https://pith.science/pith/SES3QYMWVHHT3SAAWDZCNASEIQ.json","view_paper":"https://pith.science/paper/SES3QYMW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.2984&json=true","fetch_graph":"https://pith.science/api/pith-number/SES3QYMWVHHT3SAAWDZCNASEIQ/graph.json","fetch_events":"https://pith.science/api/pith-number/SES3QYMWVHHT3SAAWDZCNASEIQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SES3QYMWVHHT3SAAWDZCNASEIQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SES3QYMWVHHT3SAAWDZCNASEIQ/action/storage_attestation","attest_author":"https://pith.science/pith/SES3QYMWVHHT3SAAWDZCNASEIQ/action/author_attestation","sign_citation":"https://pith.science/pith/SES3QYMWVHHT3SAAWDZCNASEIQ/action/citation_signature","submit_replication":"https://pith.science/pith/SES3QYMWVHHT3SAAWDZCNASEIQ/action/replication_record"}},"created_at":"2026-05-18T01:22:19.060045+00:00","updated_at":"2026-05-18T01:22:19.060045+00:00"}