17.84 (2010)

An associative algebra $A$ is said to be Calabi–Yau of dimension $d$ (for short, $\text{CY}_d$) if there is a natural isomorphism of $A$-bimodules $\text{Ext}^d_{A\text{-bimod}}(A, A \otimes A) \cong A$ and $\text{Ext}^n_{A\text{-bimod}}(A, A \otimes A) = 0$ for $n \neq d$. By Kontsevich’s theorem, the complex group algebra $\mathbb{C}G$ of the fundamental group $G$ of a 3-dimensional aspherical manifold is $\text{CY}_3$. Is every group with $\text{CY}_3$ complex group algebra residually finite?