{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:1994:37KGATG7K27ANOMHSO26I5CACP","short_pith_number":"pith:37KGATG7","schema_version":"1.0","canonical_sha256":"dfd4604cdf56be06b98793b5e4744013c66ab4848a26b3c11d87afb9090e26e3","source":{"kind":"arxiv","id":"math/9403222","version":1},"attestation_state":"computed","paper":{"title":"Hausdorff dimension and Kleinian groups","license":"","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Christopher J. Bishop, Peter Jones","submitted_at":"1994-03-22T00:00:00Z","abstract_excerpt":"Let G be a non-elementary, finitely generated Kleinian group, Lambda(G) its limit set and Omega(G) = S \\ Lambda(G) (S = the sphere) its set of discontinuity. Let delta(G) be the critical exponent for the Poincar\\'e series and let Lambda_c be the conical limit set of G. Suppose Omega_0 is a simply connected component of Omega(G). We prove that\n  (1) delta(G) = dim(Lambda_c).\n  (2) A simply connected component Omega is either a disk or dim(Omega)>1$.\n  (3) Lambda(G) is either totally disconnected, a circle or has dimension > 1,\n  (4) G is geometrically infinite iff dim(Lambda)=2.\n  (5) If G_n \\t"},"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/9403222","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"1994-03-22T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"247255b0119f97d28a2963205b280cd1092abec2166f4ecb456a7f97e8a398e5","abstract_canon_sha256":"8197445189e7cee62b910d32766b117980deab542604ec8e00454aedd3ba37a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.529921Z","signature_b64":"xt2aIjfflDOtw7wvK8AcglK86ffG6jx0qlMc6JJRZTkvfNBvvGZvb3asTSJT7Byi796UDZT6BKMcvgkX56yXBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfd4604cdf56be06b98793b5e4744013c66ab4848a26b3c11d87afb9090e26e3","last_reissued_at":"2026-05-18T01:05:51.529474Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.529474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hausdorff dimension and Kleinian groups","license":"","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Christopher J. Bishop, Peter Jones","submitted_at":"1994-03-22T00:00:00Z","abstract_excerpt":"Let G be a non-elementary, finitely generated Kleinian group, Lambda(G) its limit set and Omega(G) = S \\ Lambda(G) (S = the sphere) its set of discontinuity. Let delta(G) be the critical exponent for the Poincar\\'e series and let Lambda_c be the conical limit set of G. Suppose Omega_0 is a simply connected component of Omega(G). We prove that\n  (1) delta(G) = dim(Lambda_c).\n  (2) A simply connected component Omega is either a disk or dim(Omega)>1$.\n  (3) Lambda(G) is either totally disconnected, a circle or has dimension > 1,\n  (4) G is geometrically infinite iff dim(Lambda)=2.\n  (5) If G_n \\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9403222","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/9403222","created_at":"2026-05-18T01:05:51.529556+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/9403222v1","created_at":"2026-05-18T01:05:51.529556+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9403222","created_at":"2026-05-18T01:05:51.529556+00:00"},{"alias_kind":"pith_short_12","alias_value":"37KGATG7K27A","created_at":"2026-05-18T12:25:47.102015+00:00"},{"alias_kind":"pith_short_16","alias_value":"37KGATG7K27ANOMH","created_at":"2026-05-18T12:25:47.102015+00:00"},{"alias_kind":"pith_short_8","alias_value":"37KGATG7","created_at":"2026-05-18T12:25:47.102015+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/37KGATG7K27ANOMHSO26I5CACP","json":"https://pith.science/pith/37KGATG7K27ANOMHSO26I5CACP.json","graph_json":"https://pith.science/api/pith-number/37KGATG7K27ANOMHSO26I5CACP/graph.json","events_json":"https://pith.science/api/pith-number/37KGATG7K27ANOMHSO26I5CACP/events.json","paper":"https://pith.science/paper/37KGATG7"},"agent_actions":{"view_html":"https://pith.science/pith/37KGATG7K27ANOMHSO26I5CACP","download_json":"https://pith.science/pith/37KGATG7K27ANOMHSO26I5CACP.json","view_paper":"https://pith.science/paper/37KGATG7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/9403222&json=true","fetch_graph":"https://pith.science/api/pith-number/37KGATG7K27ANOMHSO26I5CACP/graph.json","fetch_events":"https://pith.science/api/pith-number/37KGATG7K27ANOMHSO26I5CACP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/37KGATG7K27ANOMHSO26I5CACP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/37KGATG7K27ANOMHSO26I5CACP/action/storage_attestation","attest_author":"https://pith.science/pith/37KGATG7K27ANOMHSO26I5CACP/action/author_attestation","sign_citation":"https://pith.science/pith/37KGATG7K27ANOMHSO26I5CACP/action/citation_signature","submit_replication":"https://pith.science/pith/37KGATG7K27ANOMHSO26I5CACP/action/replication_record"}},"created_at":"2026-05-18T01:05:51.529556+00:00","updated_at":"2026-05-18T01:05:51.529556+00:00"}