{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:V2O47IMRVAA3Q6NDWWA4DKKN7L","short_pith_number":"pith:V2O47IMR","schema_version":"1.0","canonical_sha256":"ae9dcfa191a801b879a3b581c1a94dfae08b94d720d3cfc2ce16df5cb9d43b8e","source":{"kind":"arxiv","id":"1102.4381","version":1},"attestation_state":"computed","paper":{"title":"Rigidity of Schottky sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Bruce Kleiner, Mario Bonk, Sergei Merenkov","submitted_at":"2011-02-22T01:45:14Z","abstract_excerpt":"We call a complement of a union of at least three disjoint (round) open balls in the unit sphere S^n a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a Mobius transformation on S^n. In the other direction we show that every Schottky set in S^2 of positive measure admits non-trivial quasisymmetric maps to other Schottky sets.\n  These results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries."},"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":"1102.4381","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-02-22T01:45:14Z","cross_cats_sorted":[],"title_canon_sha256":"3d9e2f7210a7788e2d622bf8cafd28136632b736ea1bcafb550d2babf2fdd4d4","abstract_canon_sha256":"3d91c856523160b946b7fb46e40efb9cf625abeece8926f2483c43c42151ca6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:09.099898Z","signature_b64":"wkcDxCYns3O0Eb3hASiVUF/B1/IPolQmWzvyviXvZHcpvWXUjPX+e8AZc7S6U/dE0A5Bg3ZQnnaLDW9c3mTnCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae9dcfa191a801b879a3b581c1a94dfae08b94d720d3cfc2ce16df5cb9d43b8e","last_reissued_at":"2026-05-18T04:28:09.099156Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:09.099156Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rigidity of Schottky sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Bruce Kleiner, Mario Bonk, Sergei Merenkov","submitted_at":"2011-02-22T01:45:14Z","abstract_excerpt":"We call a complement of a union of at least three disjoint (round) open balls in the unit sphere S^n a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a Mobius transformation on S^n. In the other direction we show that every Schottky set in S^2 of positive measure admits non-trivial quasisymmetric maps to other Schottky sets.\n  These results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4381","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":"1102.4381","created_at":"2026-05-18T04:28:09.099255+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.4381v1","created_at":"2026-05-18T04:28:09.099255+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.4381","created_at":"2026-05-18T04:28:09.099255+00:00"},{"alias_kind":"pith_short_12","alias_value":"V2O47IMRVAA3","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_16","alias_value":"V2O47IMRVAA3Q6ND","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_8","alias_value":"V2O47IMR","created_at":"2026-05-18T12:26:42.757692+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/V2O47IMRVAA3Q6NDWWA4DKKN7L","json":"https://pith.science/pith/V2O47IMRVAA3Q6NDWWA4DKKN7L.json","graph_json":"https://pith.science/api/pith-number/V2O47IMRVAA3Q6NDWWA4DKKN7L/graph.json","events_json":"https://pith.science/api/pith-number/V2O47IMRVAA3Q6NDWWA4DKKN7L/events.json","paper":"https://pith.science/paper/V2O47IMR"},"agent_actions":{"view_html":"https://pith.science/pith/V2O47IMRVAA3Q6NDWWA4DKKN7L","download_json":"https://pith.science/pith/V2O47IMRVAA3Q6NDWWA4DKKN7L.json","view_paper":"https://pith.science/paper/V2O47IMR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.4381&json=true","fetch_graph":"https://pith.science/api/pith-number/V2O47IMRVAA3Q6NDWWA4DKKN7L/graph.json","fetch_events":"https://pith.science/api/pith-number/V2O47IMRVAA3Q6NDWWA4DKKN7L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V2O47IMRVAA3Q6NDWWA4DKKN7L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V2O47IMRVAA3Q6NDWWA4DKKN7L/action/storage_attestation","attest_author":"https://pith.science/pith/V2O47IMRVAA3Q6NDWWA4DKKN7L/action/author_attestation","sign_citation":"https://pith.science/pith/V2O47IMRVAA3Q6NDWWA4DKKN7L/action/citation_signature","submit_replication":"https://pith.science/pith/V2O47IMRVAA3Q6NDWWA4DKKN7L/action/replication_record"}},"created_at":"2026-05-18T04:28:09.099255+00:00","updated_at":"2026-05-18T04:28:09.099255+00:00"}