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For any countably infinite group $\\Gamma$, there exists a free continuous action $\\zeta: \\Gamma \\curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $\\alpha: \\Gamma \\curvearrowright X$ on the standard Borel space, there exists an injective Borel map $\\Theta_\\alpha: X\\to C$ such that $\\Theta_\\alpha\\circ \\alpha=\\zeta \\circ \\Theta_\\alpha$. We extend our theorem for (nonfree) Borel $(\\Gamma,Z)$-actions, where $Z$ is a uniformly recurrent subgroup."},"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":"1803.05461","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-03-14T18:13:27Z","cross_cats_sorted":[],"title_canon_sha256":"166e3c4e764e0c0de3a27bebbc556301170ef5866030aa618634c7abcaf61e43","abstract_canon_sha256":"84552355d5240ea04ee1da1ae4d7e5dc71b9e01cc7cda4caf9285469e5d9f279"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:51.516146Z","signature_b64":"r62REfk9xYPzNb09DcgPH2Lq4NWBa/wmf/j7RwqgHPIfGAfOnKLLj2keu1q1mBvORkTrrjMJLhvRQ3RCNn1+Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6dc8e30d8412e900749ea4a50a5b727881c4a4fa55300efc910f5cbface8bfca","last_reissued_at":"2026-05-18T00:20:51.515615Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:51.515615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On universal continuous actions on the Cantor set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"G\\'abor Elek","submitted_at":"2018-03-14T18:13:27Z","abstract_excerpt":"Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group $\\Gamma$, there exists a free continuous action $\\zeta: \\Gamma \\curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $\\alpha: \\Gamma \\curvearrowright X$ on the standard Borel space, there exists an injective Borel map $\\Theta_\\alpha: X\\to C$ such that $\\Theta_\\alpha\\circ \\alpha=\\zeta \\circ \\Theta_\\alpha$. 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