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We show that M is commensurable with M'. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other.\n We also prove that if M is as above and G is a finitely generated"},"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":"math/0502209","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2005-02-10T13:48:45Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"5b81282182c9c76b7f281c02f91c5e0918ef0d337e9a9fb2a5ead27e8106b5e7","abstract_canon_sha256":"ab2711be962b521ec4808f51f0ae7c144954b410c3676758b4e5b51d48aea6aa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:23.757634Z","signature_b64":"Yrf6WzE8Tw+fA4QyY5JVgOPSdhbt7Ww53+7J6P+rxx5JBZGVd6ZqosD9+h3J8ORCqopSsVswoQR+MzI+cJxSAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e95080298aa9442d780f1fc53581b435c8780a6fe86af89ed359e6c911484f6","last_reissued_at":"2026-05-18T01:05:23.757143Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:23.757143Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commensurability of hyperbolic manifolds with geodesic boundary","license":"","headline":"","cross_cats":["math.DG"],"primary_cat":"math.GT","authors_text":"Roberto Frigerio","submitted_at":"2005-02-10T13:48:45Z","abstract_excerpt":"Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to the word metric). Also suppose that if n=3, then the boundaries of M and of M' are compact. We show that M is commensurable with M'. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other.\n We also prove that if M is as above and G is a finitely generated"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0502209","kind":"arxiv","version":1},"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":"math/0502209","created_at":"2026-05-18T01:05:23.757220+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0502209v1","created_at":"2026-05-18T01:05:23.757220+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0502209","created_at":"2026-05-18T01:05:23.757220+00:00"},{"alias_kind":"pith_short_12","alias_value":"J2KQQAUYVKKE","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_16","alias_value":"J2KQQAUYVKKEFV4A","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_8","alias_value":"J2KQQAUY","created_at":"2026-05-18T12:25:53.335082+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/J2KQQAUYVKKEFV4A6H6FGWA3IN","json":"https://pith.science/pith/J2KQQAUYVKKEFV4A6H6FGWA3IN.json","graph_json":"https://pith.science/api/pith-number/J2KQQAUYVKKEFV4A6H6FGWA3IN/graph.json","events_json":"https://pith.science/api/pith-number/J2KQQAUYVKKEFV4A6H6FGWA3IN/events.json","paper":"https://pith.science/paper/J2KQQAUY"},"agent_actions":{"view_html":"https://pith.science/pith/J2KQQAUYVKKEFV4A6H6FGWA3IN","download_json":"https://pith.science/pith/J2KQQAUYVKKEFV4A6H6FGWA3IN.json","view_paper":"https://pith.science/paper/J2KQQAUY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0502209&json=true","fetch_graph":"https://pith.science/api/pith-number/J2KQQAUYVKKEFV4A6H6FGWA3IN/graph.json","fetch_events":"https://pith.science/api/pith-number/J2KQQAUYVKKEFV4A6H6FGWA3IN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J2KQQAUYVKKEFV4A6H6FGWA3IN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J2KQQAUYVKKEFV4A6H6FGWA3IN/action/storage_attestation","attest_author":"https://pith.science/pith/J2KQQAUYVKKEFV4A6H6FGWA3IN/action/author_attestation","sign_citation":"https://pith.science/pith/J2KQQAUYVKKEFV4A6H6FGWA3IN/action/citation_signature","submit_replication":"https://pith.science/pith/J2KQQAUYVKKEFV4A6H6FGWA3IN/action/replication_record"}},"created_at":"2026-05-18T01:05:23.757220+00:00","updated_at":"2026-05-18T01:05:23.757220+00:00"}