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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2303.17429","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2023-03-30T14:51:32Z","cross_cats_sorted":["math.OA","math.PR"],"title_canon_sha256":"856345c920444083c882e158d091cbf4aad4bb634b1ad8bff773f8c8c7780085","abstract_canon_sha256":"93765a269cbda5e2ddfe4a4fa3f66fdd73c483caa004e74e8643492a7a4bb344"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-25T01:17:42.244096Z","signature_b64":"eHtqPb47DbKwvgqXc9VNS3blo6thizCUKsbYQHvgR3BpzQ1VbO/FFnY9bqd0MLIWVkDUtSi6cp3jvjyNTPUYDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"590bd6b430478da854fcbb64399f5816c1b4fc5965364075257fae53b4e4ec3b","last_reissued_at":"2026-06-25T01:17:42.243597Z","signature_status":"signed_v1","first_computed_at":"2026-06-25T01:17:42.243597Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\Gamma$. 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