{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:FTCNJ6NPW5ORDY53O5HZYDXBSY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3035195b654542e9ba03b6b76b7409a4496d764c6193470ee13f459c4c46ca0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-07T11:43:17Z","title_canon_sha256":"f4b9613ddd00c178ab162edff0f859dd788070b717b161a728652a43ed0d98c8"},"schema_version":"1.0","source":{"id":"1204.1619","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.1619","created_at":"2026-05-18T03:55:01Z"},{"alias_kind":"arxiv_version","alias_value":"1204.1619v2","created_at":"2026-05-18T03:55:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.1619","created_at":"2026-05-18T03:55:01Z"},{"alias_kind":"pith_short_12","alias_value":"FTCNJ6NPW5OR","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FTCNJ6NPW5ORDY53","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FTCNJ6NP","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:9dccf7f806ad8b0edf7ee88b94637b0c48c15d9d64d1a56d416e09dc5d428ebb","target":"graph","created_at":"2026-05-18T03:55:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove a Margulis' Lemma \\`a la Besson Courtois Gallot, for manifolds whose fundamental group is a nontrivial free product A*B, without 2-torsion. Moreover, if A*B is torsion-free we give a lower bound for the homotopy systole in terms of upper bounds on the diameter and the volume entropy. We also provide examples and counterexamples showing the optimality of our assumption. Finally we give two applications of this result: a finiteness theorem and a volume estimate for reducible manifolds.","authors_text":"Filippo Cerocchi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-07T11:43:17Z","title":"Margulis Lemma, entropy and free products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.1619","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fe7148d55c5b739c2836288c1f986b26faf63266ca7b1bf827fdd88478cd31cd","target":"record","created_at":"2026-05-18T03:55:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3035195b654542e9ba03b6b76b7409a4496d764c6193470ee13f459c4c46ca0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-04-07T11:43:17Z","title_canon_sha256":"f4b9613ddd00c178ab162edff0f859dd788070b717b161a728652a43ed0d98c8"},"schema_version":"1.0","source":{"id":"1204.1619","kind":"arxiv","version":2}},"canonical_sha256":"2cc4d4f9afb75d11e3bb774f9c0ee196211de8eb958db28e25346b2060e54470","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2cc4d4f9afb75d11e3bb774f9c0ee196211de8eb958db28e25346b2060e54470","first_computed_at":"2026-05-18T03:55:01.430064Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:55:01.430064Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rfPkxcE8BiaMXpw3I2jRzsiJW2Naf3ppdutDsrP6bykDB2knHwF2lzAe/8dE3GX7O32BkfBZTyf9/Ep4L8wtCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:55:01.430529Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.1619","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fe7148d55c5b739c2836288c1f986b26faf63266ca7b1bf827fdd88478cd31cd","sha256:9dccf7f806ad8b0edf7ee88b94637b0c48c15d9d64d1a56d416e09dc5d428ebb"],"state_sha256":"2037e3f6b3596b580e780079073cc0405e9e488c07494edf8b60cb108fa9f687"}