{"paper":{"title":"Coxeter groups, hyperbolic cubes, and acute triangulations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Genevieve S. Walsh, Sang-hyun Kim","submitted_at":"2013-06-25T16:37:23Z","abstract_excerpt":"Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an abstract triangulation of $\\mathbb{S}^2$ can be realized as an acute triangulation exactly when it satisfies a combinatorial condition called \"flag no-square\". We also study generalizations of this result to other angle bounds, other planar surfaces and other dimensions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6025","kind":"arxiv","version":3},"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"}