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 = Z<sup>2</sup> we will see that the two cases |X| < c and |X| ≥ c must be treated very differently. We derive necessary conditions for a group action with |X| < c 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| ≥ c we give further constructions of compact Hausdorff topologies for which the group action is continuous.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:432253 |
| Date | January 2006 |
| Creators | Suabedissen, Rolf |
| Contributors | Collins, Peter J. ; Knight, Robin W. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:ca186b79-b4ea-423f-a697-5ed47b4672f9 |
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