{"paper":{"title":"Profinite rigidity of simple closed curves in surface groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Xiaoyu Xu, Zhongzi Wang","submitted_at":"2026-07-17T17:27:08Z","abstract_excerpt":"This paper establishes a new characterization of simple closed curves on a closed orientable surface. Let $\\Gamma$ be the fundamental group of a closed orientable surface. We prove that if an element $g\\in\\Gamma$ has the same possible images as a given simple closed curve $\\gamma\\in \\Gamma$ under epimorphisms from $\\Gamma$ to every finite group, then $g$ belongs to the $\\mathrm{Aut}(\\Gamma)$-orbit of $\\gamma$, i.e. $g$ is itself a simple closed curve with the same topological type as $\\gamma$. Consequently, the set of simple closed curves in $\\Gamma$ is closed in the profinite topology of $\\Ga"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16147","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.16147/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}