Let $G$ be a discrete group that acts on a contractible finite dimensional $G$-complex $X$ with the following properties:

- $X/G$ is compact (i.e. the action is
*cocompact*) - Each stabilizer $G_\sigma$ admits a cocompact action on a contractible finite dimensional $G_\sigma$-complex with finite stabilizers

**Question:** Is there a finite dimensional contractible $G$-complex $Y$
with *finite* stabilizers such that $Y/G$ is compact ?

The question can be stated more conceptually by help of the following definition: Let $\mathscr{F}$ be the class of all finite groups and define the class $K_i\mathscr{F},\;i\ge 0$ inductively by

- $K_0\mathscr{F} := \mathscr{F}$
- $K_i\mathscr{F}$ includes all groups $G$ that admit a finite dimensional contractible $G$-complex $X$ such that (1) $X/G$ is compact and (2) the stabilizers are in $K_{i-1}\mathscr{F}$.

Then the question is equivalent to

**Question:** Is $K_1\mathscr{F} \subsetneqq K_2\mathscr{F}$ ?

Remark: If condition (1) is dropped, we get the classical Kropholler classes $H_i\mathscr{F}$. There (among many other results) $H_1\mathscr{F} \subsetneqq H_2\mathscr{F}$ is known: For example, the free abelian group of countably infinite rank belongs to $H_2\mathscr{F}\setminus H_1\mathscr{F}$.