{"paper":{"title":"Hyperbolic extensions of free groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Samuel J. Taylor, Spencer Dowdall","submitted_at":"2014-06-10T14:26:58Z","abstract_excerpt":"Given a finitely generated subgroup $\\Gamma \\le \\mathrm{Out}(\\mathbb{F})$ of the outer automorphism group of the rank $r$ free group $\\mathbb{F} = F_r$, there is a corresponding free group extension $1 \\to \\mathbb{F} \\to E_{\\Gamma} \\to \\Gamma \\to 1$. We give sufficient conditions for when the extension $E_{\\Gamma}$ is hyperbolic. In particular, we show that if all infinite order elements of $\\Gamma$ are atoroidal and the action of $\\Gamma$ on the free factor complex of $\\mathbb{F}$ has a quasi-isometric orbit map, then $E_{\\Gamma}$ is hyperbolic. As an application, we produce examples of hyper"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.2567","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"}