  A Cantor action is a minimal equicontinuous action of a countably generated
group G on a Cantor space X. Such actions are also called generalized odometers
in the literature. In this work, we introduce two new conjugacy invariants for
Cantor actions, the stabilizer limit group and the centralizer limit group. An
action is wild if the stabilizer limit group is an increasing sequence of
stabilizer groups without bound, and otherwise is said to be stable if this
group chain is bounded. For Cantor actions by a finitely generated group G, we
prove that stable actions satisfy a rigidity principle, and furthermore show
that the wild property is an invariant of the continuous orbit equivalence
class of the action.
  A Cantor action is said to be dynamically wild if it is wild, and the
centralizer limit group is a proper subgroup of the stabilizer limit group.
This property is also a conjugacy invariant, and we show that a Cantor action
with a non-Hausdorff element must be dynamically wild. We then give examples of
wild Cantor actions with non-Hausdorff elements, using recursive methods from
Geometric Group Theory to define actions on the boundaries of trees.
