{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:NYRWQYPDOESLITVVGPPKREM7DO","short_pith_number":"pith:NYRWQYPD","canonical_record":{"source":{"id":"1810.00657","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-10-01T12:31:52Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"f62f920c9bdbe8343c2aba24163508a37c4386a6e96b7546fd8e75656db3a5cf","abstract_canon_sha256":"ad79e5fa6478019335f661a6fb7af20476634a4d87f0dcd575ae35060e9ccb37"},"schema_version":"1.0"},"canonical_sha256":"6e236861e37124b44eb533dea8919f1bb93bd725c29f4129fe7995ca146bcd97","source":{"kind":"arxiv","id":"1810.00657","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.00657","created_at":"2026-05-17T23:42:48Z"},{"alias_kind":"arxiv_version","alias_value":"1810.00657v4","created_at":"2026-05-17T23:42:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.00657","created_at":"2026-05-17T23:42:48Z"},{"alias_kind":"pith_short_12","alias_value":"NYRWQYPDOESL","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NYRWQYPDOESLITVV","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NYRWQYPD","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:NYRWQYPDOESLITVVGPPKREM7DO","target":"record","payload":{"canonical_record":{"source":{"id":"1810.00657","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-10-01T12:31:52Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"f62f920c9bdbe8343c2aba24163508a37c4386a6e96b7546fd8e75656db3a5cf","abstract_canon_sha256":"ad79e5fa6478019335f661a6fb7af20476634a4d87f0dcd575ae35060e9ccb37"},"schema_version":"1.0"},"canonical_sha256":"6e236861e37124b44eb533dea8919f1bb93bd725c29f4129fe7995ca146bcd97","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:48.836038Z","signature_b64":"iEfecpTzf537T2xCp281GoqerObxeHjcYwXFcEmzaY/Z5oZNSzKEN9G6wDZpTxMbakPvEzX/YFnzQ6BJ/UGZCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e236861e37124b44eb533dea8919f1bb93bd725c29f4129fe7995ca146bcd97","last_reissued_at":"2026-05-17T23:42:48.835578Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:48.835578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.00657","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gXVRueub3NSgNSMRZuK8QtDqjr1a5LRi+2xb8JIaIbMPHNPSJgFm42hufh9IXWanGtMBwM3GIzNcwIfQucdHAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T12:10:34.012828Z"},"content_sha256":"3faffff6c9f3cac786ea00df555434ef1b5cc6afce68480ad993a4e84801b300","schema_version":"1.0","event_id":"sha256:3faffff6c9f3cac786ea00df555434ef1b5cc6afce68480ad993a4e84801b300"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:NYRWQYPDOESLITVVGPPKREM7DO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On discreteness of subgroups of quaternionic hyperbolic isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.GT","authors_text":"Devendra Tiwari, Krishnendu Gongopadhyay, Mukund Madhav Mishra","submitted_at":"2018-10-01T12:31:52Z","abstract_excerpt":"Let ${{\\bf H}_{\\mathbb H}}^n$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group ${\\rm{Sp}}(n,1)$ acts by the isometries of ${{\\bf H}_{\\mathbb H}}^n$. A subgroup $G$ of ${\\rm {Sp}}(n,1)$ is called \\emph{Zariski dense} if it does not fix a point on ${{\\bf H}_{\\mathbb H}}^n \\cup \\partial {{\\bf H}_{\\mathbb H}}^n$ and neither it preserves a totally geodesic subspace of ${{{\\bf H}}_{\\mathbb H}}^n$. We prove that a Zariski dense subgroup $G$ of ${\\rm{ Sp}}(n,1)$ is discrete if for every loxodromic element $g \\in G$ the two generator subgroup $\\langle f, g f g^{-1} \\rangle$ is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.00657","kind":"arxiv","version":4},"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":"2026-05-17T23:42:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OF9FucssqWP5GJF/rx/1E3FPFiwm1xVwa34r1P4QZCoc7S9O8bCjcUhsP2qcPTFaCQ6wSpDsyKRtb3yGq4BtBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T12:10:34.013178Z"},"content_sha256":"92f4c7ddf2b45b58658fb9be30f8cdf631fa3274f7aa485c50034750a55bc655","schema_version":"1.0","event_id":"sha256:92f4c7ddf2b45b58658fb9be30f8cdf631fa3274f7aa485c50034750a55bc655"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO/bundle.json","state_url":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NYRWQYPDOESLITVVGPPKREM7DO/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":"2026-06-22T12:10:34Z","links":{"resolver":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO","bundle":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO/bundle.json","state":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NYRWQYPDOESLITVVGPPKREM7DO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NYRWQYPDOESLITVVGPPKREM7DO","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":"ad79e5fa6478019335f661a6fb7af20476634a4d87f0dcd575ae35060e9ccb37","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-10-01T12:31:52Z","title_canon_sha256":"f62f920c9bdbe8343c2aba24163508a37c4386a6e96b7546fd8e75656db3a5cf"},"schema_version":"1.0","source":{"id":"1810.00657","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.00657","created_at":"2026-05-17T23:42:48Z"},{"alias_kind":"arxiv_version","alias_value":"1810.00657v4","created_at":"2026-05-17T23:42:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.00657","created_at":"2026-05-17T23:42:48Z"},{"alias_kind":"pith_short_12","alias_value":"NYRWQYPDOESL","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NYRWQYPDOESLITVV","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NYRWQYPD","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:92f4c7ddf2b45b58658fb9be30f8cdf631fa3274f7aa485c50034750a55bc655","target":"graph","created_at":"2026-05-17T23:42:48Z","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":"Let ${{\\bf H}_{\\mathbb H}}^n$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group ${\\rm{Sp}}(n,1)$ acts by the isometries of ${{\\bf H}_{\\mathbb H}}^n$. A subgroup $G$ of ${\\rm {Sp}}(n,1)$ is called \\emph{Zariski dense} if it does not fix a point on ${{\\bf H}_{\\mathbb H}}^n \\cup \\partial {{\\bf H}_{\\mathbb H}}^n$ and neither it preserves a totally geodesic subspace of ${{{\\bf H}}_{\\mathbb H}}^n$. We prove that a Zariski dense subgroup $G$ of ${\\rm{ Sp}}(n,1)$ is discrete if for every loxodromic element $g \\in G$ the two generator subgroup $\\langle f, g f g^{-1} \\rangle$ is","authors_text":"Devendra Tiwari, Krishnendu Gongopadhyay, Mukund Madhav Mishra","cross_cats":["math.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-10-01T12:31:52Z","title":"On discreteness of subgroups of quaternionic hyperbolic isometries"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.00657","kind":"arxiv","version":4},"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":"sha256:3faffff6c9f3cac786ea00df555434ef1b5cc6afce68480ad993a4e84801b300","target":"record","created_at":"2026-05-17T23:42:48Z","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":"ad79e5fa6478019335f661a6fb7af20476634a4d87f0dcd575ae35060e9ccb37","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-10-01T12:31:52Z","title_canon_sha256":"f62f920c9bdbe8343c2aba24163508a37c4386a6e96b7546fd8e75656db3a5cf"},"schema_version":"1.0","source":{"id":"1810.00657","kind":"arxiv","version":4}},"canonical_sha256":"6e236861e37124b44eb533dea8919f1bb93bd725c29f4129fe7995ca146bcd97","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6e236861e37124b44eb533dea8919f1bb93bd725c29f4129fe7995ca146bcd97","first_computed_at":"2026-05-17T23:42:48.835578Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:48.835578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iEfecpTzf537T2xCp281GoqerObxeHjcYwXFcEmzaY/Z5oZNSzKEN9G6wDZpTxMbakPvEzX/YFnzQ6BJ/UGZCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:48.836038Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.00657","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3faffff6c9f3cac786ea00df555434ef1b5cc6afce68480ad993a4e84801b300","sha256:92f4c7ddf2b45b58658fb9be30f8cdf631fa3274f7aa485c50034750a55bc655"],"state_sha256":"4141de077b1e839184ec90b5b873bb04d43af1d3ad974aefe762e717d3f3dd02"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7kNw7AZ/ttGcY9NoEEcf/rXef1fUSFK/qnpzhydOc0WVomVJ27Ws4YVwr2rixVr1cY6HV9OhdlZ7nzcmajS9BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T12:10:34.015116Z","bundle_sha256":"cac124b1ec491c25ef390333c31dcc3eeab17109c0c657c40f08ddae34466d75"}}