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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"},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1810.00657","created_at":"2026-05-17T23:42:48.835649+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.00657v4","created_at":"2026-05-17T23:42:48.835649+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.00657","created_at":"2026-05-17T23:42:48.835649+00:00"},{"alias_kind":"pith_short_12","alias_value":"NYRWQYPDOESL","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"NYRWQYPDOESLITVV","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"NYRWQYPD","created_at":"2026-05-18T12:32:40.477152+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/NYRWQYPDOESLITVVGPPKREM7DO","json":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO.json","graph_json":"https://pith.science/api/pith-number/NYRWQYPDOESLITVVGPPKREM7DO/graph.json","events_json":"https://pith.science/api/pith-number/NYRWQYPDOESLITVVGPPKREM7DO/events.json","paper":"https://pith.science/paper/NYRWQYPD"},"agent_actions":{"view_html":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO","download_json":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO.json","view_paper":"https://pith.science/paper/NYRWQYPD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.00657&json=true","fetch_graph":"https://pith.science/api/pith-number/NYRWQYPDOESLITVVGPPKREM7DO/graph.json","fetch_events":"https://pith.science/api/pith-number/NYRWQYPDOESLITVVGPPKREM7DO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO/action/storage_attestation","attest_author":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO/action/author_attestation","sign_citation":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO/action/citation_signature","submit_replication":"https://pith.science/pith/NYRWQYPDOESLITVVGPPKREM7DO/action/replication_record"}},"created_at":"2026-05-17T23:42:48.835649+00:00","updated_at":"2026-05-17T23:42:48.835649+00:00"}