{"paper":{"title":"Invariant percolation and measured theory of nonamenable groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.OA","math.PR"],"primary_cat":"math.GR","authors_text":"Cyril Houdayer","submitted_at":"2011-06-27T09:34:54Z","abstract_excerpt":"Using percolation techniques, Gaboriau and Lyons recently proved that every countable, discrete, nonamenable group $\\Gamma$ contains measurably the free group $\\mathbf F_2$ on two generators: there exists a probability measure-preserving, essentially free, ergodic action of $\\mathbf F_2$ on $([0, 1]^\\Gamma, \\lambda^\\Gamma)$ such that almost every $\\Gamma$-orbit of the Bernoulli shift splits into $\\mathbf F_2$-orbits. A combination of this result and works of Ioana and Epstein shows that every countable, discrete, nonamenable group admits uncountably many non-orbit equivalent actions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5337","kind":"arxiv","version":1},"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"}