{"paper":{"title":"Lamplighter groups and von Neumann's continuous regular rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RA","authors_text":"Gabor Elek","submitted_at":"2014-02-22T10:50:12Z","abstract_excerpt":"Let $\\Gamma$ be a discrete group. Following Linnell and Schick one can define a continuous ring $c(\\Gamma)$ associated with $\\Gamma$. They proved that if the Atiyah Conjecture holds for a torsion-free group $\\Gamma$, then $c(\\Gamma)$ is a skew field. Also, if $\\Gamma$ has torsion and the Strong Atiyah\n  Conjecture holds for $\\Gamma$, then $c(\\Gamma)$ is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group $\\Gamma=Z_2\\wr Z$. It is known that\n  $C(Z_2\\wr Z)$ does not even have a classical ring of quotients. Our main result is that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5499","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"}