Extensions of group actions and the Hilbert-Smith Conjecture

The Hilbert-Smith Conjecture proposes that every effective compact group action on a compact manifold is a Lie group. The conjecture is the generalization of Hilbert's fifth problem, and it is still open for dimensions 4 and higher. It is well known that the conjecture is equivalent to postulating that there is no effective action of a p-adic group on a compact manifold.;We explore several well-known examples of free and effective p-adic group actions on spaces that are not manifolds. We provide constructions for other actions on similar spaces by the group of p-adic numbers and other pro-finite groups. We prove that for any Peano continuum admitting an effective p-adic action, the continuum can be equivariantly partitioned. While the quotient map of the action is not generally a covering map, we extend many standard results on covering maps to it.;We also present a different approach to the Hilbert-Smith conjecture by looking internally at the space under the action of the group. We show that free p-adic actions on the space of irrationals are unique up to conjugation. We also show that should a counter-example to the Hilbert-Smith conjecture exist, the counter-example would be an extension of this unique free p-adic group action on the space of irrationals. This motivates the investigation of compact extensions of compact metric group actions on separable metric spaces. While we show that there is always an extension of a p-adic action to some metric compactification, a free action does not necessarily extend to a free action. We give sufficient conditions to guarantee an extension of a group action to a given compactification. We present examples of group actions failing to extend without those conditions.;The conjecture is translated into terms of the ring of continuous functions on the space of irrationals. We give an equivalent version of the conjecture in these terms, and explore how the new setting facilitates our investigation of extending actions and answering the conjecture. We give sufficient conditions for ensuring that being able to extend each homeomorphism in a group action to a compactification will have the group action extend.