{"paper":{"title":"Classifying $GL(2,\\mathbb Z) \\ltimes \\mathbb Z^{2}$-orbits by subgroups of $\\mathbb R$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Daniele Mundici","submitted_at":"2014-01-15T19:12:02Z","abstract_excerpt":"Let $\\mathcal G_2$ denote the affine group $GL(2,\\mathbb Z) \\ltimes \\mathbb Z^{2}$. For every point $x=(x_1,x_2) \\in \\R2$ let $\\orb(x)=\\{y\\in\\R2\\mid y=\\gamma(x)$ for some $\\gamma \\in \\mathcal{G}_2 \\}$. Let $G_{x}$ be the subgroup of the additive group $\\mathbb R$ generated by $x_1,x_2, 1$. If $\\rank(G_x)\\in \\{1,3\\}$ then $\\orb(x)=\\{y\\in\\R2\\mid G_y=G_x\\}$. If $\\rank(G_x)=2$, knowledge of $G_x$ is not sufficient in general to uniquely recover $\\orb(x)$: rather, $G_x$ classifies precisely $\\max(1,\\phi(d)/2)$ different orbits, where $d$ is the denominator of the smallest positive nonzero rational "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.3708","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":""},"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"}