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Consequently for free abelian groups of infinite rank there is no sequence F_n along which the RET holds, and in many finitely generated groups, including the discrete Heisenberg group and the free group on d>1 generators, there is no (sub)sequence of balls, in the standard generators, along which the RET holds.\n  On the other hand, in groups wit"},"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":"1208.1002","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-08-05T11:53:29Z","cross_cats_sorted":[],"title_canon_sha256":"4874cde83acc3d9891c5ef638fbcbe1729493130ab628f18c49b38dcb68ad6ec","abstract_canon_sha256":"a098fc2803ac70874ebde51aceb8335a84cd45f00c502aa5ba09116079ebb8e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:15.276380Z","signature_b64":"zQzfuMBbrUiCnY/6CZs40XrlJa2/ZKMWbvmduxzM3dWlr4OPZ2vyv3D4nd3lBjZGS6tghYiHhPZjBX0UTGb1CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15f031d382c2c0634582dd1ef93c4762a5059f0a7f30fd2cc4fa3f2f997f34c0","last_reissued_at":"2026-05-18T02:42:15.275643Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:15.275643Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the ratio ergodic theorem for group actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Michael Hochman","submitted_at":"2012-08-05T11:53:29Z","abstract_excerpt":"We study the ratio ergodic theorem (RET) of Hopf for group actions. 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