{"paper":{"title":"On freeness of the random fundamental group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR","math.PR"],"primary_cat":"math.AT","authors_text":"Andrew Newman","submitted_at":"2016-01-27T19:54:36Z","abstract_excerpt":"Let $Y(n, p)$ denote the probability space of random 2-dimensional simplicial complexes in the Linial--Meshulam model, and let $Y \\sim Y(n, p)$ denote a random complex chosen according to this distribution. In a paper of Cohen, Costa, Farber, and Kappeler, it is shown that for $p = o(1/n)$ with high probability $\\pi_1(Y)$ is free. Following that, a paper of Costa and Farber shows that for values of $p$ which satisfy $3/n < p \\ll n^{-46/47}$, with high probability $\\pi_1(Y)$ is not free. Here we improve on both of these results to show that there are explicit constants $\\gamma_2 < c_2 < 3$, so "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07520","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"}