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If $A$ has eigenvalues $\\lambda_1 = \\lambda e^{i\\varphi},$ $\\lambda_2 = e^{-2i\\varphi}$, $\\lambda_3 = \\lambda^{-1}e^{i\\varphi}$, we prove that $\\Gamma$ is conjugate in ${\\rm SU}(2,1)$ to a subgroup of ${\\rm SU}(2,1,\\mathbb{Q}(\\Gamma,\\lambda)),$ where $\\mathbb{Q}(\\Gamma, \\lambda)$ is the field generated by the trace field $\\mathbb{Q}(\\Gamma)$ of $\\Gamma$ and $\\lambda$. It follows from this that if $\\Gamma$ is an irreducible subgroup of ${\\rm SU}(2,1)$ such that the trace fiel"},"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":"1303.1701","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-03-07T14:26:02Z","cross_cats_sorted":[],"title_canon_sha256":"9ca231aa1c862bd5bb41ee6b36fc9cc73690a5d4021ec73532a685c25b00e624","abstract_canon_sha256":"dc5fe2409824d40f3c6ca1c3e88012acc77c2083463829361929d93e2a0f0a42"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:31:33.394132Z","signature_b64":"1Tmx16azt8sPrBtFua+oCVRs53dFpcPU904BuAYKkS5AOjtIvbKWbWyg/rmjgpU1RGVAon2f25ShnAcfPcKFAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"739f4cade1ea2436c493e0993187746cc150074b0cc7e6628c7a24088c8933c8","last_reissued_at":"2026-05-18T03:31:33.393670Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:31:33.393670Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on trace fields of complex hyperbolic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Heleno Cunha, Nikolay Gusevskii","submitted_at":"2013-03-07T14:26:02Z","abstract_excerpt":"We show that if $\\Gamma$ is an irreducible subgroup of ${\\rm SU}(2,1)$, then $\\Gamma$ contains a loxodromic element $A$. If $A$ has eigenvalues $\\lambda_1 = \\lambda e^{i\\varphi},$ $\\lambda_2 = e^{-2i\\varphi}$, $\\lambda_3 = \\lambda^{-1}e^{i\\varphi}$, we prove that $\\Gamma$ is conjugate in ${\\rm SU}(2,1)$ to a subgroup of ${\\rm SU}(2,1,\\mathbb{Q}(\\Gamma,\\lambda)),$ where $\\mathbb{Q}(\\Gamma, \\lambda)$ is the field generated by the trace field $\\mathbb{Q}(\\Gamma)$ of $\\Gamma$ and $\\lambda$. 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