Topologizing Group Actions

This thesis is centered on the following question: Given an abstract group action by an Abelian group G on a set X, when is there a compact Hausdorff topology on X such that the group action is continuous? If such a topology exists, we call the group action compact-realizable. We show that if G is a locally-compact group, a necessary condition for a G-action to be compact-realizable, is that the image of X under the stabilizer map must be a compact subspace of the collection of closed subgroups of G equipped with the co-compact topology. We apply this result to give a complete characterization for the case when G is a compact Abelian group in terms of the existence of continuous compact Hausdorff pre-images of a certain topological space associated with the group action. If G is not compact, we will show that the necessary condition is not sufficient. Together with various examples, we then present a general two-stage method of construction for compact Hausdorff topologies for ℝ-actions. For discrete groups, the necessary condition above turns out to be not very strong. In the case of G = ℤ2 we will see that the two cases |X| < and |X| ≥ must be treated very differently. We derive necessary conditions for a group action with |X| < to be compact-realizable by constructing particularly nice open partitions of the space X. We then use symbolic dynamics together with some generic constructions to obtain a partial converse in this case. If |X| ≥ we give further constructions of compact Hausdorff topologies for which the group action is continuous.