{"paper":{"title":"On the procongruence completion of the Teichm\\\"uller modular group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.NT"],"primary_cat":"math.AG","authors_text":"Marco Boggi","submitted_at":"2009-10-22T12:39:04Z","abstract_excerpt":"For $2g-2+n>0$, the Teichm\\\"uller modular group $\\Gamma_{g,n}$ of a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$ is the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $\\Pi_{g,n}$ be the fundamental group of $S_{g,n}$, with a given base point, and $\\hat{\\Pi}_{g,n}$ its profinite completion. There is then a natural faithful representation $\\Gamma_{g,n}\\hookrightarrow Out(\\hat{\\Pi}_{g,n})$. The procongruence completion $\\check{\\Gamma}_{g,n}$ of the Teichm\\\"uller group is define"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.4305","kind":"arxiv","version":5},"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"}