{"paper":{"title":"Non-arithmeticity of length spectra of subgroups of mapping class groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Every non-elementary subgroup of the mapping class group has a non-arithmetic Teichmüller length spectrum.","cross_cats":["math.DS","math.GR"],"primary_cat":"math.GT","authors_text":"Dongryul M. Kim, Inhyeok Choi","submitted_at":"2026-05-13T06:39:25Z","abstract_excerpt":"In this paper, we prove that every non-elementary subgroup of the mapping class group of a surface has non-arithmetic Teichm\\\"uller length spectrum. Namely, Teichm\\\"uller translation lengths of its pseudo-Anosov elements generate a dense additive subgroup of $\\mathbb{R}$. We prove this by introducing the notion of cross-ratios on $\\mathcal{MF}$ and $\\mathcal{PMF}$, and studying its geometric and dynamical properties, despite the lack of negatively curved features of the Teichm\\\"uller space nor the conformal geometry on $\\mathcal{PMF}$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"every non-elementary subgroup of the mapping class group of a surface has non-arithmetic Teichmüller length spectrum. Namely, Teichmüller translation lengths of its pseudo-Anosov elements generate a dense additive subgroup of R.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The cross-ratios on MF and PMF satisfy the geometric and dynamical properties needed to force density of the length spectrum, despite the absence of negative curvature or conformal structure.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Non-elementary subgroups of mapping class groups have non-arithmetic Teichmüller length spectra, shown via new cross-ratios on measured foliations and projective measured foliations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Every non-elementary subgroup of the mapping class group has a non-arithmetic Teichmüller length spectrum.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"817dc90931c21c79f7580c84dad5e09778a9d0088a5316e697a6f995d76e2fa8"},"source":{"id":"2605.13064","kind":"arxiv","version":1},"verdict":{"id":"ab4a4e5b-31a1-437e-b0c9-b5f15ccf4886","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T01:49:05.610809Z","strongest_claim":"every non-elementary subgroup of the mapping class group of a surface has non-arithmetic Teichmüller length spectrum. Namely, Teichmüller translation lengths of its pseudo-Anosov elements generate a dense additive subgroup of R.","one_line_summary":"Non-elementary subgroups of mapping class groups have non-arithmetic Teichmüller length spectra, shown via new cross-ratios on measured foliations and projective measured foliations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The cross-ratios on MF and PMF satisfy the geometric and dynamical properties needed to force density of the length spectrum, despite the absence of negative curvature or conformal structure.","pith_extraction_headline":"Every non-elementary subgroup of the mapping class group has a non-arithmetic Teichmüller length spectrum."},"references":{"count":29,"sample":[{"doi":"","year":2024,"title":"Effective mapping class group dynamics III : counting filling closed curves on surfaces","work_id":"8ba20acb-20cc-4569-ba15-1c022a2ee0f3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"Propri\\'et\\'es asymptotiques des groupes lin\\'eaires","work_id":"f79a60b8-4fcf-4058-b679-6241f986fdfb","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1978,"title":"An extremal problem for quasiconformal mappings and a theorem by T hurston","work_id":"f2ca80c5-0885-4dbc-80ed-ec3704947d6c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1995,"title":"M. Bourdon. Structure conforme au bord et flot g\\' e od\\' e sique d'un CAT (-1) -espace. Enseign. Math. (2) , 41(1-2):63--102, 1995","work_id":"6230750a-e161-4476-bdd4-f44d39d3b789","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Inhyeok Choi and Dongryul M. Kim. Invariant measures on the space of measured laminations for subgroups of mapping class group. arXiv preprint arXiv:2510.23256 , 2025","work_id":"1dd7c443-feaa-4096-9504-f2b8b29b11f7","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":29,"snapshot_sha256":"270be4c13f2886a6ab4eca958137e62ebc0831c83b5dab6df294f19950df5693","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"760c9249c6c8c4f97420568adf67772fb97cef2628afa856217c6f23ad0ace99"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}