{"paper":{"title":"Measure theory in the geometry of $GL(n,\\mathbb Z) \\ltimes \\mathbb Z^{n}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Daniele Mundici","submitted_at":"2011-02-04T12:51:55Z","abstract_excerpt":"The $n$-dimensional affine group over the integers is the group $\\mathcal G_n$ of all affinities on $\\mathbb R^{n}$ which leave the lattice $ \\mathbb Z^{n}$ invariant. $\\mathcal G_n$ yields a geometry in the classical sense of the Erlangen Program. In this paper we construct a $\\mathcal G_n$-invariant measure on rational polyhedra in $\\mathbb R^n$, i.e., finite unions of simplexes with rational vertices in $\\mathbb R^n$, and prove its uniqueness. Our main tool is given by the Morelli-W{\\l}odarczyk factorization of birational toric maps in blow-ups and blow-downs (solution of the weak Oda conje"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0897","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"}