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However if $G$ has one end the number of ends of $H$ may be larger; we observe cases where this occurs for $\\C$ the class of finite $p$-groups.\n  We produce a monomorphism of groups $u:H\\hookrightarrow G$ such that: either $G$ is hyperbolic but not"},"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":"1205.0271","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-05-01T22:31:08Z","cross_cats_sorted":[],"title_canon_sha256":"1528cd57b2ac18f223559162d2725ce10ef975aa00a501c97571a8cf4d1b35a6","abstract_canon_sha256":"4e167d7f580eba334f556a30dea3b4515a5187cfa3fd19f3d4c9258f6ddee43e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:57:39.213374Z","signature_b64":"Fs8OHszOIcAHoC25yu/iEvj7kWRO2R/PfDCbQ1g17PQCWR/i13FoFJddFCNLaIumuH5TvDFOeqQNgIdVx9BNCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c6f034aa0e1daa3467159511bb4528f233475e7e0d1ad9c4d14db8f8e267850","last_reissued_at":"2026-05-18T01:57:39.212807Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:57:39.212807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Detecting ends of residually finite groups in profinite completions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Owen Cotton-Barratt","submitted_at":"2012-05-01T22:31:08Z","abstract_excerpt":"Let $\\C$ be a variety of finite groups. 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