{"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"}