Lamplighter groups and von Neumann's continuous regular rings

2014/02/22 by Gábor Elek, Elek, Gabor · 2 citations
Engineering · Mathematics · #16A30 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1402.5499

openalex publication_date 2014/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ be a discrete group. Following Linnell and Schick one can define a continuous ring c(Γ) associated with Γ. They proved that if the Atiyah Conjecture holds for a torsion-free group Γ, then c(Γ) is a skew field. Also, if Γ has torsion and the Strong Atiyah Conjecture holds for Γ, then c(Γ) is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group Γ=Z₂\wr Z. It is known that C(Z₂\wr Z) does not even have a classical ring of quotients. Our main result is that if H is amenable, then c(Z₂\wr H) is isomorphic to a continuous ring constructed by John von Neumann in the 1930's.

Cited by

• Projective simplicity and Bergman's property for unit groups of continuous rings
• Geometric properties of unit groups of von Neumann's continuous rings

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