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

TL;DR

This paper confirms that for sofic groups, twisted $L^2$-Betti numbers are equal to the usual $L^2$-Betti numbers scaled by the dimension of the twisting representation, answering a question posed by Wolfgang Lück.

Contribution

The paper proves that twisted $L^2$-Betti numbers of sofic groups equal the usual $L^2$-Betti numbers multiplied by the representation dimension, confirming Lück's conjecture.

Findings

Twisted $L^2$-Betti numbers equal scaled usual $L^2$-Betti numbers for sofic groups

Confirmation of Lück's conjecture in the context of sofic groups

Advancement in understanding the behavior of $L^2$-Betti numbers under twisting

Abstract

Wolfang L\"uck asked if twisted $L^2$-Betti numbers of a group are equal to the usual $L^2$-Betti numbers rescaled by the dimension of the twisting representation. We confirm this for sofic groups.