{"paper":{"title":"Quantitative Bounded Distance Theorem and Margulis' Lemma for Z^n actions with applications to homology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.MG","authors_text":"Andrea Sambusetti, Filippo Cerocchi","submitted_at":"2014-12-19T20:33:17Z","abstract_excerpt":"We consider the stable norm associated to a discrete, torsionless abelian group of isometries $\\Gamma \\cong \\mathbb{Z}^n$ of a geodesic space $(X,d)$. We show that the difference between the stable norm $\\| \\;\\, \\|_{st}$ and the distance $d$ is bounded by a constant only depending on the rank $n$ and on upper bounds for the diameter of $\\bar X=\\Gamma \\backslash X$ and the asymptotic volume $\\omega(\\Gamma, d)$. We also prove that the upper bound on the asymptotic volume is equivalent to a lower bound for the stable systole of the action of $\\Gamma$ on $(X,d)$; for this, we establish a Lemma \\`a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6516","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"}