  Using fixed-point-free group actions, we set up a scheme to define nested
classes of groups indexed over ordinals. Restricting to cellular actions on
CW-complexes, we find new classes as well as new characterizations for some
well-known classes, such as virtually polycyclic groups. We generalize
properties of the virtual cohomological dimension of a group to groups with
jump (co)homology and prove that a core subclass of a new class of groups has
jump (co)homology.
