{"paper":{"title":"Homogeneous length functions on groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.GT","math.MG"],"primary_cat":"math.GR","authors_text":"D.H.J. Polymath","submitted_at":"2018-01-11T18:12:21Z","abstract_excerpt":"A pseudo-length function defined on an arbitrary group $G = (G,\\cdot,e, (\\,)^{-1})$ is a map $\\ell: G \\to [0,+\\infty)$ obeying $\\ell(e)=0$, the symmetry property $\\ell(x^{-1}) = \\ell(x)$, and the triangle inequality $\\ell(xy) \\leqslant \\ell(x) + \\ell(y)$ for all $x,y \\in G$. We consider pseudo-length functions which saturate the triangle inequality whenever $x=y$, or equivalently those that are homogeneous in the sense that $\\ell(x^n) = n\\,\\ell(x)$ for all $n\\in\\mathbb{N}$. We show that this implies that $\\ell([x,y])=0$ for all $x,y \\in G$. This leads to a classification of such pseudo-length "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03908","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"}