Kropholler's conjecture on finite-dimensional models for proper actions

Let $\mathfrak{F}$ be the class of finite groups. A group $G$ belongs to ${\scriptstyle \mathbf H}_{1}\mathfrak{F}$ if it admits a finite-dimensional contractible $G$-$\operatorname{CW}$-complex whose cell stabilisers lie in $\mathfrak{F}$. A model for $E_{\operatorname{\mathfrak{F}}}G$ is a proper $G$-$\operatorname{CW}$-complex whose $K$-fixed point subcomplex is contractible for every finite subgroup $K$ of $G$. Kropholler's conjecture. Every ${\scriptstyle \mathbf H}_{1}\mathfrak{F}$-group $G$ admits a finite-dimensional model for $E_{\operatorname{\mathfrak{F}}}G$. This conjecture asks whether the geometric finiteness condition defining ${\scriptstyle \mathbf H}_{1}\mathfrak{F}$ guarantees a finite-dimensional classifying space for proper actions, and was open for almost 20 years at the time of the source.