{"paper":{"title":"Digraphs and cycle polynomials for free-by-cyclic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Eriko Hironaka, Kasra Rafi, Yael Algom-Kfir","submitted_at":"2013-10-28T18:57:56Z","abstract_excerpt":"Let $\\phi \\in \\mbox{Out}(F_n)$ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism $\\phi$ determines a free-by-cyclic group $\\Gamma=F_n \\rtimes_\\phi \\mathbb Z,$ and a homomorphism $\\alpha \\in H^1(\\Gamma; \\mathbb Z)$. By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, $\\alpha$ has an open cone neighborhood $\\mathcal A$ in $H^1(\\Gamma;\\mathbb R)$ whose integral points correspond to other fibrations of $\\Gamma$ whose associated outer automorphisms are themselves representable by expanding irreducible trai"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.7533","kind":"arxiv","version":2},"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"}