{"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. Bridson","submitted_at":"2010-07-15T15:56:17Z","abstract_excerpt":"We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F_n) \\to Out(F_m). 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2598","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}