Groups possessing extensive hierarchical decompositions

Kropholler's class of groups is the smallest class of groups which contains all finite groups and is closed under the following operator: whenever $G$ admits a finite-dimensional contractible $G$-CW-complex in which all stabilizer groups are in the class, then $G$ is itself in the class. Kropholler's class admits a hierarchical structure, i.e., a natural filtration indexed by the ordinals. For example, stage 0 of the hierarchy is the class of all finite groups, and stage 1 contains all groups of finite virtual cohomological dimension. We show that for each countable ordinal $\alpha$, there is a countable group that is in Kropholler's class which does not appear until the $\alpha+1$st stage of the hierarchy. Previously this was known only for $\alpha= 0$, 1 and 2. The groups that we construct contain torsion. We also review the construction of a torsion-free group that lies in the third stage of the hierarchy.