Twisted \(L^2\)-Betti numbers for sofic groups (Q6614912)

Let \(G\) be a group and \(\mathrm{rk}_{G}\) the von Neumann Sylvester matrix rank function of the group algebra \(\mathbb{C}[G]\). Although \(\mathrm{rk}_{G}\) is defined in an analytic way, it may be characterized algebraically for some groups (for locally indicable groups, see [\textit{A. Jaikin-Zapirain} and \textit{D. López-Álvarez}, Math. Ann. 376, No. 3--4, 1741--1793 (2020; Zbl 1481.20093)]).\N\NLet \(\sigma: G \rightarrow \mathrm{GL}_{k}(\mathbb{C})\) be a representation and define \(\widetilde{\sigma}: \mathbb{C}[G] \rightarrow \mathrm{Mat}_{k}(\mathbb{C}[G])\) by sending \(g \in G\) to \(\sigma(g) g\) and then extending by linearity. A modified version of a conjecture proposed by \textit{W. Lück} [J. Topol. Anal. 10, No. 4, 723--816 (2018; Zbl 1411.57037)] asserts that for every matrix \(A \in \mathrm{Mat}_{m \times n}(\mathbb{C}[G])\) the identity \(\mathrm{rk}_{G}(\widetilde{\sigma}(A))=k \cdot \mathrm{rk}_{G}(A)\) is valid (Lück twisted conjecture). Lück's conjecture was proved for torsion-free elementary amenable groups [Lück, loc. cit.] and for locally indicable groups [\textit{D. Kielak} and \textit{B. Sun}, Math. Ann. 390, No. 3, 3567--3619 (2024; Zbl 07932387)].\N\NThe main result of the paper under review is Theorem 1: The Lück twisted conjecture holds for sofic groups.\N\NSince amenable and residually finite groups are sofic, Theorem 1 provides a large class of examples in which the Lück twisted conjecture is valid.