{"paper":{"title":"Fixed points of nilpotent actions on ${\\mathbb S}^{2}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Javier Rib\\'on","submitted_at":"2012-08-22T14:32:44Z","abstract_excerpt":"We prove that a nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\\mathbb S}^{2}$ has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\\mathbb R}^{2}$ preserving a compact set has a global fixed point. These results generalize theorems of Franks, Handel and Parwani for the abelian case.\n  We show that a nilpotent subgroup of orientation preserving $C^{1}$ diffeomorphisms of ${\\mathbb S}^{2}$ that has a finite orbit of odd cardinality also has a global fixed point."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4510","kind":"arxiv","version":4},"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"}