L²-Betti numbers arising from the lamplighter group

2020/05/26 by Ara, Pere, Claramunt, Joan
#16E50 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Primary 20C07 #Rings and Algebras (math.RA) #Secondary 20F65

paper · doi:10.48550/arxiv.2005.12814

Abstract

We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute ℓ²-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational ℓ²-Betti numbers arising from the lamplighter group algebra K[ℤ₂ \wr ℤ], being K a subfield of the complex numbers closed under complex conjugation. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational ℓ²-Betti numbers from the algebras ℚ[ℤ_n \wr ℤ], where n ≥ 2 is a natural number. We also apply the techniques developed to the (generalized) odometer algebra O(n), where n is a supernatural number. We compute its *-regular closure, and this allows us to fully characterize the set of ℓ²-Betti numbers arising from O(n).

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