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Then the actions of $\\Gamma_1$ and $\\Gamma_2$ coinduced from $T_1$ and $T_2$ are orbit-equivalent. As an application, it is shown that if $\\Gamma$ is a free group, then all nontrivial Bernoulli shifts over $\\Gamma$ are orbit-equivalent."},"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":"0906.4573","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-06-24T22:08:50Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"164002a8943f613b7c118d2ec3d6479e3ba1ee3566b9e90b99724660201ce42b","abstract_canon_sha256":"f9efd64da844056ff4dbc672b54f1b17c48b89f0674a8a52b3debaf277e0bccb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:55.351735Z","signature_b64":"TyLHSUXeYaqpFxpAj9g60YlbrZ372pvDam6yNW+d2TKThUrYD9SXSFP+ejQoYcUCFOy9BRBOUHc87AQR47wQCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e987ed71ead0121c4079381f1ab589bf9e722765ef7ed56c1aeaae1640b3697d","last_reissued_at":"2026-05-18T02:24:55.350967Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:55.350967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Orbit equivalence, coinduced actions and free products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.DS","authors_text":"Lewis Bowen","submitted_at":"2009-06-24T22:08:50Z","abstract_excerpt":"The following result is proven. Let $G_1 \\cc^{T_1} (X_1,\\mu_1)$ and $G_2 \\cc^{T_2} (X_2,\\mu_2)$ be orbit-equivalent, essentially free, probability measure preserving actions of countable groups $G_1$ and $G_2$. Let $H$ be any countable group. For $i=1,2$, let $\\Gamma_i = G_i *H$ be the free product. Then the actions of $\\Gamma_1$ and $\\Gamma_2$ coinduced from $T_1$ and $T_2$ are orbit-equivalent. As an application, it is shown that if $\\Gamma$ is a free group, then all nontrivial Bernoulli shifts over $\\Gamma$ are orbit-equivalent."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.4573","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0906.4573","created_at":"2026-05-18T02:24:55.351077+00:00"},{"alias_kind":"arxiv_version","alias_value":"0906.4573v3","created_at":"2026-05-18T02:24:55.351077+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.4573","created_at":"2026-05-18T02:24:55.351077+00:00"},{"alias_kind":"pith_short_12","alias_value":"5GD624PK2AJB","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_16","alias_value":"5GD624PK2AJBYQDZ","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_8","alias_value":"5GD624PK","created_at":"2026-05-18T12:25:58.837520+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5GD624PK2AJBYQDZHAPRVNMJX6","json":"https://pith.science/pith/5GD624PK2AJBYQDZHAPRVNMJX6.json","graph_json":"https://pith.science/api/pith-number/5GD624PK2AJBYQDZHAPRVNMJX6/graph.json","events_json":"https://pith.science/api/pith-number/5GD624PK2AJBYQDZHAPRVNMJX6/events.json","paper":"https://pith.science/paper/5GD624PK"},"agent_actions":{"view_html":"https://pith.science/pith/5GD624PK2AJBYQDZHAPRVNMJX6","download_json":"https://pith.science/pith/5GD624PK2AJBYQDZHAPRVNMJX6.json","view_paper":"https://pith.science/paper/5GD624PK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0906.4573&json=true","fetch_graph":"https://pith.science/api/pith-number/5GD624PK2AJBYQDZHAPRVNMJX6/graph.json","fetch_events":"https://pith.science/api/pith-number/5GD624PK2AJBYQDZHAPRVNMJX6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5GD624PK2AJBYQDZHAPRVNMJX6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5GD624PK2AJBYQDZHAPRVNMJX6/action/storage_attestation","attest_author":"https://pith.science/pith/5GD624PK2AJBYQDZHAPRVNMJX6/action/author_attestation","sign_citation":"https://pith.science/pith/5GD624PK2AJBYQDZHAPRVNMJX6/action/citation_signature","submit_replication":"https://pith.science/pith/5GD624PK2AJBYQDZHAPRVNMJX6/action/replication_record"}},"created_at":"2026-05-18T02:24:55.351077+00:00","updated_at":"2026-05-18T02:24:55.351077+00:00"}