{"paper":{"title":"Haagerup property and group-invariant percolation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OA","math.PR"],"primary_cat":"math.GR","authors_text":"Chiranjib Mukherjee, Konstantin Recke","submitted_at":"2023-03-30T14:51:32Z","abstract_excerpt":"Let $\\mathcal G$ be the Cayley graph of a finitely generated, infinite group $\\Gamma$. We show that $\\Gamma$ has the Haagerup property if and only if for every $\\alpha<1$, there is a $\\Gamma$-invariant bond percolation $\\mathbb P$ on $\\mathcal G$ with $\\mathbb E[\\mathrm{deg}_{\\omega}(g)]>\\alpha\\mathrm{deg}_{\\mathcal G}(g)$ for every vertex $g$ and with the two-point function $\\tau(g,h)=\\mathbb P\\big[g\\leftrightarrow h\\big]$ vanishing as $d(g,h)\\to\\infty$. On the other hand, we show that $\\Gamma$ has Kazhdan's property (T) if and only if there exists a threshold $\\alpha^*<1$ such that for every"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.17429","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.17429/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}