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The subgroup $\\Gamma$ determines an extension $E_\\Gamma$ of $F_N$, and the main theorem of Dowdall--Taylor \\cite{DT1} states that in this situation $E_\\Gamma$ is hyperbolic if and only if $\\Gamma$ is purely atoroidal.\n  Here,"},"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":"1506.06974","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-06-23T12:51:18Z","cross_cats_sorted":["math.DS","math.GT"],"title_canon_sha256":"98dc45d76ff7b8939a628dcd8af9643ce7ae17911305735a81386dbc383d8933","abstract_canon_sha256":"3faf8dfa5c5c5f17371133bf5d0e3743f85187e4e17871c2c7d1d489a391c36c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:24.732668Z","signature_b64":"cX7tzOwR3SthiPLnIc88iWG2ITixCw+RRYQeWV4kvZck2n2SK+NQpVvsZvIaVnvobw1Utsi2Uyr510kkdqKIBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41e06cc89f50719043de89325701b139e19efb81897e3aa71ed5b4fcbd017a8a","last_reissued_at":"2026-05-18T01:24:24.732036Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:24.732036Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cannon-Thurston maps for hyperbolic free group extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GT"],"primary_cat":"math.GR","authors_text":"Ilya Kapovich, Samuel J. 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