{"paper":{"title":"Groups with classifiable actions on the line","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A finitely generated group G lies outside class C but is amenable precisely when Thompson's group F is amenable.","cross_cats":["math.DS","math.LO"],"primary_cat":"math.GR","authors_text":"Joaqu\\'in Brum, Mart\\'in Gilabert Vio, Nicol\\'as Matte Bon","submitted_at":"2026-05-13T12:00:43Z","abstract_excerpt":"We motivate and study the class $\\mathcal{C}$ of countable groups $G$ such that the conjugacy relation between minimal actions of $G$ on $\\mathbb{R}$ by orientation-preserving homeomorphisms is smooth -- that is, admits a Borel transversal. No example of amenable group outside of $\\mathcal{C}$ is known. We show a number of stability properties of $\\mathcal{C}$ under group-theoretic operations and that $\\mathcal{C}$ contains all finitely generated groups of piecewise affine homeomorphisms of the interval. We exhibit a finitely generated group $G$ that is not in $\\mathcal{C}$, such that $G$ is a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We exhibit a finitely generated group G that is not in C, such that G is amenable if and only if Thompson's group F is amenable.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the smoothness of the conjugacy relation (existence of a Borel transversal) is the right notion for 'classifiable' actions and that the explicit construction of G satisfies all the claimed dynamical and group-theoretic properties without hidden assumptions on the actions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Introduces class C of groups with smooth conjugacy for minimal line actions, proves stability and inclusion of piecewise affine groups, and exhibits a finitely generated G not in C with amenability equivalent to Thompson's F.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A finitely generated group G lies outside class C but is amenable precisely when Thompson's group F is amenable.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"dea27e1b2819624ca9eb25e204758986aeebdb061430d0fcc03a721d2a4ef171"},"source":{"id":"2605.13406","kind":"arxiv","version":1},"verdict":{"id":"c82a778d-4d5b-4ef4-a155-603bc2c05954","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:31:29.690395Z","strongest_claim":"We exhibit a finitely generated group G that is not in C, such that G is amenable if and only if Thompson's group F is amenable.","one_line_summary":"Introduces class C of groups with smooth conjugacy for minimal line actions, proves stability and inclusion of piecewise affine groups, and exhibits a finitely generated G not in C with amenability equivalent to Thompson's F.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the smoothness of the conjugacy relation (existence of a Borel transversal) is the right notion for 'classifiable' actions and that the explicit construction of G satisfies all the claimed dynamical and group-theoretic properties without hidden assumptions on the actions.","pith_extraction_headline":"A finitely generated group G lies outside class C but is amenable precisely when Thompson's group F is amenable."},"references":{"count":49,"sample":[{"doi":"","year":1984,"title":"V. A. Antonov, Modeling of processes of cyclic evolution type. S ynchronization by a random signal , Vestnik Leningradskogo Universiteta. Matematika, Mekhanika, Astronomiya (1984), no. vyp. 2, pp. 67-","work_id":"f22b0f5b-97e3-487f-9138-1b14d38b01f2","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.2307/2371787","year":1946,"title":"Richard Arens, Topologies for homeomorphism groups https://doi.org/10.2307/2371787, American Journal of Mathematics 68 (1946), pp. 593--610","work_id":"5132d3ca-4cfa-44a7-9ad8-4252c8655ce0","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1940,"title":"Arsenin, Sur les projections de certains ensembles mesurables B , Doklady Akademii Nauk SSSR 27 (1940), pp","work_id":"fee048a3-804f-4235-8045-d05fa94c7b5d","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.4171/jca/49","year":2021,"title":"Brin, and Justin Tatch Moore, https://doi.org/10.4171/jca/49 Complexity among the finitely generated subgroups of T hompson's group , Journal of Combinatorial Algebra 5 (2021), no","work_id":"61d095fd-45b9-44a7-8c34-c82923bcd1c0","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1017/cbo9780511735264","year":1996,"title":"Kechris, https://doi.org/10.1017/CBO9780511735264 The descriptive set theory of P olish group actions , London Mathematical Society Lecture Note Series, vol","work_id":"7d149df0-2f14-4292-b0e1-782d178c8023","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":49,"snapshot_sha256":"46b9c2135f56c36216d9b960e4282d0f843c0a3c3aea7b3965d7f986d160fc58","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"a5dd0d8f59c10650c024cdbdc42ed85c57d12678783a888fbdf71e45c867062b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}