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.