{"paper":{"title":"Remarks on minimal sets and conjectures of Cassels, Swinnerton-Dyer, and Margulis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Barak Weiss, Jinpeng An","submitted_at":"2012-12-02T15:09:34Z","abstract_excerpt":"We prove that a hypothesis of Cassels, Swinnerton-Dyer, recast by Margulis as statement on the action of the diagonal group $A$ on the space of unimodular lattices, is equivalent to several assertions about minimal sets for this action. More generally, for a maximal $\\mathbb{R}$-diagonalizable subgroup $A$ of a reductive group $G$ and a lattice $\\Gamma$ in $G$, we give a sufficient condition for a compact $A$-minimal subset $Y$ of $G/\\Gamma$ to be of a simple form, which is also necessary if $G$ is $\\mathbb{R}$-split. We also show that the stabilizer of $Y$ has no nontrivial connected unipoten"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0216","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"}