Twisted L 2 $L^2$ -Betti numbers for sofic groups

IF 0.8 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-04-24 DOI:10.1112/blms.13050

Jan Boschheidgen, Andrei Jaikin-Zapirain

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Abstract

For a given group $G$ , Wolfgang Lück asked whether twisting a chain complex of finitely generated free $C [G]$ -modules with a finite-dimensional complex representation $V$ of $G$ before passing to the $L^{2}$ -completion has no other effect on $L^{2}$ -Betti numbers than a scaling by the factor ${dim}_{C} V$ . The purpose of the article is to answer this question affirmatively provided $G$ is sofic.