  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.
