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We conjecture that if $G$ is a profinite group in which every element admits a sink that is a procyclic subgroup, then $G$ is procyclic-by-(locally nilpotent). We prove the conjecture in two cases -- when $G$ is a finite group, or a soluble pro-$p$ group."},"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":"1905.07494","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-05-17T22:33:52Z","cross_cats_sorted":[],"title_canon_sha256":"3a1e229e7c424c7b09507ce965aca3a88744fd61e1ab51ee9423208d48c8dfef","abstract_canon_sha256":"9b6e943d51b844f85b1830681d01b4649a55d86d5d55cdd2439ccbf26c2fc949"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:49.560028Z","signature_b64":"eywt26TFvnvvWyQLQ5CKLgXN9d8fB/rnwVMT3Idefe+90wJlqAUG/lcXkapTWT+wDqQBfrQZCG8xYYYU5dlzBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6adb3c8ce45a6bf0230099b687985c965908f125bcc5be2340d9aa7543b129b3","last_reissued_at":"2026-05-17T23:45:49.559382Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:49.559382Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On groups in which Engel sinks are cyclic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Cristina Acciarri, Pavel Shumyatsky","submitted_at":"2019-05-17T22:33:52Z","abstract_excerpt":"For an element $g$ of a group $G$, an Engel sink is a subset $\\mathcal{E}(g)$ such that for every $ x\\in G $ all sufficiently long commutators $ [x,g,g,\\ldots,g] $ belong to $\\mathcal{E}(g)$. 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