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In particular, if $G \\curvearrowright (X, \\mu)$ has infinite entropy, then the action $G \\curvearrowright (Y, \\nu)$ does not admit any finite generating partition. On the other hand, we prove that if $G$ is a countable non-amenable group then there exists a finite integer $n$ with the following property: for every probability-measure-preserving action $G \\c"},"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":"1311.0738","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-04T15:47:02Z","cross_cats_sorted":["math.GR","math.LO"],"title_canon_sha256":"054e9645112834b299ec2a8758bb0bf85869652ab9cff82ace9b9e1c0faa2456","abstract_canon_sha256":"4971ba4da861cbfc096da5d6027cf8c7154496ecc83f3d8774bef0e82aab0c35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:24.631229Z","signature_b64":"WAN662MoFD2Od5DjTxbGzEO/q41T+UyjssXB8Zuq0XXUUE0pp7dU8i/J4CfzeRrJ0dsk51aB8NYQyjLIDxFAAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95ac890d027a22de92a559e20bb6cd9378917757b6a5fa9c91003599e2fad420","last_reissued_at":"2026-05-18T02:48:24.630782Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:24.630782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Every action of a non-amenable group is the factor of a small action","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.LO"],"primary_cat":"math.DS","authors_text":"Brandon Seward","submitted_at":"2013-11-04T15:47:02Z","abstract_excerpt":"It is well known that if $G$ is a countable amenable group and $G \\curvearrowright (Y, \\nu)$ factors onto $G \\curvearrowright (X, \\mu)$, then the entropy of the first action must be greater than or equal to the entropy of the second action. 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