{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:AB2K5UZTRGDZIXX2PCJXNFCY3L","short_pith_number":"pith:AB2K5UZT","canonical_record":{"source":{"id":"1506.08113","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-26T15:17:17Z","cross_cats_sorted":[],"title_canon_sha256":"474faf77501467dff58af2a7048709e8ff5c79cfe0ccd1c6e1fda0cc6082f3ca","abstract_canon_sha256":"aac9ffe06b5f738b3a12727438132e1347f61cd2d5351b62354c8439298ae1c0"},"schema_version":"1.0"},"canonical_sha256":"0074aed3338987945efa7893769458dac9af0196ecd40ebe969e4a55c6d1c873","source":{"kind":"arxiv","id":"1506.08113","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.08113","created_at":"2026-05-18T01:12:30Z"},{"alias_kind":"arxiv_version","alias_value":"1506.08113v5","created_at":"2026-05-18T01:12:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.08113","created_at":"2026-05-18T01:12:30Z"},{"alias_kind":"pith_short_12","alias_value":"AB2K5UZTRGDZ","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"AB2K5UZTRGDZIXX2","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"AB2K5UZT","created_at":"2026-05-18T12:29:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:AB2K5UZTRGDZIXX2PCJXNFCY3L","target":"record","payload":{"canonical_record":{"source":{"id":"1506.08113","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-26T15:17:17Z","cross_cats_sorted":[],"title_canon_sha256":"474faf77501467dff58af2a7048709e8ff5c79cfe0ccd1c6e1fda0cc6082f3ca","abstract_canon_sha256":"aac9ffe06b5f738b3a12727438132e1347f61cd2d5351b62354c8439298ae1c0"},"schema_version":"1.0"},"canonical_sha256":"0074aed3338987945efa7893769458dac9af0196ecd40ebe969e4a55c6d1c873","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:30.772205Z","signature_b64":"89oVzIWx4vIYWpG+1/FuOETjoXts3zb+uncwOlM9LNseEFehstUmlmkG76wNLMIatBrU+nFCnODxhGPz5mVNCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0074aed3338987945efa7893769458dac9af0196ecd40ebe969e4a55c6d1c873","last_reissued_at":"2026-05-18T01:12:30.771881Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:30.771881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.08113","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"suHQLFjVoYmCzp4naogSG97gBoJgwn6tjB/Cz71xXhsb/NGGs2HuwjaelbY2e/avdKh4YgGNjJhqAlg8/zGwDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:13:36.611712Z"},"content_sha256":"0450d984661e630973f8523f82aa6bfd5565c1dd3fd64c38c341012fa88eab6c","schema_version":"1.0","event_id":"sha256:0450d984661e630973f8523f82aa6bfd5565c1dd3fd64c38c341012fa88eab6c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:AB2K5UZTRGDZIXX2PCJXNFCY3L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The limit set for discrete complex hyperbolic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Angel Cano, Bingyuan Liu, Marlon M. L\\'opez","submitted_at":"2015-06-26T15:17:17Z","abstract_excerpt":"Given a discrete subgroup $\\Gamma$ of $PU(1,n)$ it acts by isometries on the unit complex ball $\\Bbb{H}^n_{\\Bbb{C}}$, in this setting a lot of work has been done in order to understand the action of the group. However when we look at the action of $\\Gamma$ on all of $ \\Bbb{P}^n_{\\Bbb{C}}$ little or nothing is known, in this paper study the action in the whole projective space and we are able to show that its equicontinuity agree with its Kulkarni discontuity set. Morever, in the non-elementary case, this set turns out to be the largest open set on which the group acts properly and discontinuou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.08113","kind":"arxiv","version":5},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PEqMPHKGiO2xjqf1IG2mizgiPO1zwcbS+h/kiTvD7gHOoy0Q1k5AnQJoQk+5BY4av48M4v1RTEo2WDGVxvjwCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:13:36.612394Z"},"content_sha256":"721d929219546049a6558be7e598f9f0fb3fa845a07c48df2a451d4004b49975","schema_version":"1.0","event_id":"sha256:721d929219546049a6558be7e598f9f0fb3fa845a07c48df2a451d4004b49975"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AB2K5UZTRGDZIXX2PCJXNFCY3L/bundle.json","state_url":"https://pith.science/pith/AB2K5UZTRGDZIXX2PCJXNFCY3L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AB2K5UZTRGDZIXX2PCJXNFCY3L/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T22:13:36Z","links":{"resolver":"https://pith.science/pith/AB2K5UZTRGDZIXX2PCJXNFCY3L","bundle":"https://pith.science/pith/AB2K5UZTRGDZIXX2PCJXNFCY3L/bundle.json","state":"https://pith.science/pith/AB2K5UZTRGDZIXX2PCJXNFCY3L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AB2K5UZTRGDZIXX2PCJXNFCY3L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:AB2K5UZTRGDZIXX2PCJXNFCY3L","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":"aac9ffe06b5f738b3a12727438132e1347f61cd2d5351b62354c8439298ae1c0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-26T15:17:17Z","title_canon_sha256":"474faf77501467dff58af2a7048709e8ff5c79cfe0ccd1c6e1fda0cc6082f3ca"},"schema_version":"1.0","source":{"id":"1506.08113","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.08113","created_at":"2026-05-18T01:12:30Z"},{"alias_kind":"arxiv_version","alias_value":"1506.08113v5","created_at":"2026-05-18T01:12:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.08113","created_at":"2026-05-18T01:12:30Z"},{"alias_kind":"pith_short_12","alias_value":"AB2K5UZTRGDZ","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"AB2K5UZTRGDZIXX2","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"AB2K5UZT","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:721d929219546049a6558be7e598f9f0fb3fa845a07c48df2a451d4004b49975","target":"graph","created_at":"2026-05-18T01:12:30Z","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":"Given a discrete subgroup $\\Gamma$ of $PU(1,n)$ it acts by isometries on the unit complex ball $\\Bbb{H}^n_{\\Bbb{C}}$, in this setting a lot of work has been done in order to understand the action of the group. However when we look at the action of $\\Gamma$ on all of $ \\Bbb{P}^n_{\\Bbb{C}}$ little or nothing is known, in this paper study the action in the whole projective space and we are able to show that its equicontinuity agree with its Kulkarni discontuity set. Morever, in the non-elementary case, this set turns out to be the largest open set on which the group acts properly and discontinuou","authors_text":"Angel Cano, Bingyuan Liu, Marlon M. L\\'opez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-26T15:17:17Z","title":"The limit set for discrete complex hyperbolic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.08113","kind":"arxiv","version":5},"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":"sha256:0450d984661e630973f8523f82aa6bfd5565c1dd3fd64c38c341012fa88eab6c","target":"record","created_at":"2026-05-18T01:12:30Z","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":"aac9ffe06b5f738b3a12727438132e1347f61cd2d5351b62354c8439298ae1c0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-26T15:17:17Z","title_canon_sha256":"474faf77501467dff58af2a7048709e8ff5c79cfe0ccd1c6e1fda0cc6082f3ca"},"schema_version":"1.0","source":{"id":"1506.08113","kind":"arxiv","version":5}},"canonical_sha256":"0074aed3338987945efa7893769458dac9af0196ecd40ebe969e4a55c6d1c873","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0074aed3338987945efa7893769458dac9af0196ecd40ebe969e4a55c6d1c873","first_computed_at":"2026-05-18T01:12:30.771881Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:30.771881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"89oVzIWx4vIYWpG+1/FuOETjoXts3zb+uncwOlM9LNseEFehstUmlmkG76wNLMIatBrU+nFCnODxhGPz5mVNCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:30.772205Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.08113","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0450d984661e630973f8523f82aa6bfd5565c1dd3fd64c38c341012fa88eab6c","sha256:721d929219546049a6558be7e598f9f0fb3fa845a07c48df2a451d4004b49975"],"state_sha256":"507fd776a628b61ee53d6ca6e28a757b9af405059d03c8d2d7a3f564d822a0b1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"le9Li7N4PdMjwqJob+Hp0Gux4UGLlgSawVn5aThECp1IsY1zmgMJHMemaQ7NcCYnN7x26X8yiLIe1MVt9iUjBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T22:13:36.616341Z","bundle_sha256":"ea51f37adbd52e1f7120d24438af0fe5f6765380dda5b1a96ae48dfbb4f7dafe"}}