Group algebra criteria for vanishing of cohomology (Q2204346)

Let \(\Gamma\) be a group acting on a contractible simplicial complex with finite stabilizers. In this paper, the authors give criteria for vanishing of cohomology groups of \(\Gamma\) with unitary representations. As it is well known that for a finitely generated group \(G\) all first cohomology groups of \(G\) with unitary representations vanish if and only if \(G\) has Kazhdan's property (T). For any \(n \geq 1\), according to [\textit{M. De Chiffre} et al., Forum Math. Sigma 8, Paper No. e18, 37 p. (2020; Zbl 1456.22002)], we say that a group \(G\) is \(n\)-Kazhdan if all \(n\)-th cohomology groups of \(G\) with unitary representations vanish. For a finite generated group, \(1\)-Kazhdan is equivalent to Kazhdan's property (T). In [loc. cit.], Chiffre et al. showed that \(2\)-Kazhdan is equivalent to Frobenius-stable for a finite presented group. Furthermore they gave short expositions of remarkable results and recent developments for the vanishing of all cohomology groups with unitary representations. In the present paper, the authors give a criterion for \(\Gamma\) to be \(n\)-Kazhdan in terms of the rational group algebra of \(\Gamma\) using \textit{N. Ozawa}'s description for property (T) [J. Inst. Math. Jussieu 15, No. 1, 85--90 (2016; Zbl 1336.22008)].