{"paper":{"title":"Normal generation and $\\ell^2$-betti numbers of groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.OA"],"primary_cat":"math.GR","authors_text":"A. Thom, D. Osin","submitted_at":"2011-08-11T14:24:22Z","abstract_excerpt":"The \\emph{normal rank} of a group is the minimal number of elements whose normal closure coincides with the group. We study the relation between the normal rank of a group and its first $\\ell^2$-Betti number and conjecture that inequality $\\beta_1^{(2)}(G)$ does not exceed normal rank minus 1 for torsion free groups. The conjecture is proved for limits of left-orderable amenable groups. On the other hand, for every $n\\ge 2$ and every $\\e>0$, we give an example of a simple group $Q$ (with torsion) such that $\\beta_1^{(2)}(Q) \\geq n-1-\\epsilon$. These groups also provide examples of simple group"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.2411","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"}