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We prove that if n > 8 is even and n \\neq m \\leq 2n, or n is odd and n \\neq m \\leq 2n - 2, then all such homomorphisms have finite image; in fact they factor through det: Out(F_n) \\to Z/2. In contrast, if m = r^n(n - 1) + 1 with r coprime to (n - 1), then there exists an embedding Out(F_n) \\to Out(F_m). In order to prove this last statement, we determine when the action of Out(F_n) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with a"},"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":"1007.2598","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-07-15T15:56:17Z","cross_cats_sorted":[],"title_canon_sha256":"edf5924ada732d8d58cc600edf020581cad64455ce42793f47e2ccf26460476c","abstract_canon_sha256":"8cb281c6289063b7430a093fec8c593a14067892fb01741551f317333e4b5076"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:40.937732Z","signature_b64":"mQb8zOL0dGv59PchG35RkSccC2WDF9cHptzS9e6t13xnBV6PZ5946cLgxYy7WVib11NgufqvvdkWTUsNaHh/AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c97a105b3d445b528e2715df59c8837ba07ac7814a68baf2df95b0b58793dbf","last_reissued_at":"2026-05-18T04:16:40.937045Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:40.937045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Abelian covers of graphs and maps between outer automorphism groups of free groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Karen Vogtmann, Martin R. 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