{"paper":{"title":"Random triangular groups at density 1/3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.GR","authors_text":"Jacek \\'Swi\\c{a}tkowski, Sylwia Antoniuk, Tomasz {\\L}uczak","submitted_at":"2013-08-27T13:32:02Z","abstract_excerpt":"Let \\Gamma(n,p) denote the binomial model of a random triangular group. We show that there exist constants c, C > 0 such that if p <= c/n^2, then a.a.s. \\Gamma(n,p) is free and if p >= C log n/n^2 then a.a.s. \\Gamma(n,p) has Kazhdan's property (T). Furthermore, we show that there exist constants C',c' > 0 such that if C'/n^2 <= p <= c' log n/n^2, then a.a.s. \\Gamma(n,p) is neither free nor has Kazhdan's property (T)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5867","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"}