{"paper":{"title":"Reducing systems for very small trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Patrick Reynolds","submitted_at":"2012-11-14T18:40:53Z","abstract_excerpt":"We study very small trees from the point of view of reducing systems of free factors, which are analogues of reducing systems of curves for a surface lamination; a non-trivial, proper free factor $F \\leq \\FN$ reduces $T$ if and only if $F$ acts on some subtree of $T$ with dense orbits. We characterize those trees, called arational, which do not admit a reduction by any free factor: $T$ is arational if and only if either $T$ is free and indecomposable or $T$ is dual to a surface with one boundary component equipped with an arational measured foliation. To complement this result, we establish so"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3378","kind":"arxiv","version":1},"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"}