{"paper":{"title":"On Arboreal Galois Representations of Rational Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.RT"],"primary_cat":"math.NT","authors_text":"Ashvin Swaminathan","submitted_at":"2014-07-25T19:25:45Z","abstract_excerpt":"The action of the absolute Galois group $\\text{Gal}(K^{\\text{ksep}}/K)$ of a global field $K$ on a tree $T(\\phi, \\alpha)$ of iterated preimages of $\\alpha \\in \\mathbb{P}^1(K)$ under $\\phi \\in K(x)$ with $\\text{deg}(\\phi) \\geq 2$ induces a homomorphism $\\rho: \\text{Gal}(K^{\\text{ksep}}/K) \\to \\text{Aut}(T(\\phi, \\alpha))$, which is called an arboreal Galois representation. In this paper, we address a number of questions posed by Jones and Manes about the size of the group $G(\\phi,\\alpha) := \\text{im} \\rho = \\underset{\\leftarrow n}\\lim\\text{Gal}(K(\\phi^{-n}(\\alpha))/K)$. Specifically, we consider"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7012","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"}