{"paper":{"title":"On cubic action of a rank one group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Matthias Gr\\\"uninger","submitted_at":"2011-06-12T13:37:53Z","abstract_excerpt":"We consider a rank one group $G = \\langle A,B \\rangle $ which acts cubically on a module $V$, this means $[V,A,A,A] =0$ but $[V,G,G,G] \\ne 0$. We have to distinguish whether the group $A_0 :=C_A([V,A]) \\cap C_A(V/C_V(A))$ is trivial or not. We show that if $A_0$ is trivial, $G$ is a rank one group associated to a quadratic Jordan division algebra. If $A_0$ is not trivial (which is always the case if $A$ is not abelian), then $A_0$ defines a subgroup $G_0$ of $G$ which acts quadratically on $V$. We will call $G_0$ the \\textit{quadratic kernel} of $G$. By a result of Timmesfeld we have $G_0 \\con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.2310","kind":"arxiv","version":3},"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"}