{"paper":{"title":"Super-approximation, I: p-adic semisimple case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GR","authors_text":"Alireza Salehi Golsefidy","submitted_at":"2016-02-01T06:51:00Z","abstract_excerpt":"Let $k$ be a number field, $\\Omega$ be a finite symmetric subset of $\\mathbb{GL}_{n_0}(k)$, and $\\Gamma=\\langle \\Omega\\rangle$. Let \\[ C(\\Gamma):=\\{\\mathfrak{p}\\in V_f(k)|\\hspace{1mm} \\Gamma \\text{is a bounded subgroup of} \\mathbb{GL}_{n_0}(k_{\\mathfrak{p}})\\}, \\] and $\\Gamma_{\\mathfrak{p}}$ be the closure of $\\Gamma$ in $\\mathbb{GL}_{n_0}(k_{\\mathfrak{p}})$. Assuming that the Zariski-closure of $\\Gamma$ is semisimple, we prove that the family of left translation actions $\\{\\Gamma\\curvearrowright \\Gamma_{\\mathfrak{p}}\\}_{\\mathfrak{p}\\in C(\\Gamma)}$ has {\\em uniform spectral gap}.\n  As a coroll"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00403","kind":"arxiv","version":2},"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"}