On vanishing of $L^{2}$-Betti numbers for groups

Abstract

We show that if a group $G$ admits a finite dimensional contractible $G$-CW-complex $X$ then the vanishing of the $L^2$-Betti numbers for all stabilizers $G_\sigma$ of $X$ determines that of the $L^2$-Betti numbers for $G$. We also give a relation among the $L^2$-Euler characteristics for $X$ as a $G$-CW-complex and those for $X$ as a $G_\sigma$-CW-complex under certain assumptions. Finally, we present a new class of groups satisfying the Chatterji-Mislin conjecture which amounts to putting Brown's formula within the framework of $L^2$-homology.