Profinite Group Actions
Posted by John Baez
I have three questions. I have some guesses about the answers, so don’t think I’m completely clueless. But I’m clueless enough that I’d prefer to just give the questions, not my guesses.
Question 1. Given a topological space , does every action of on a finite set extend to a continuous action of the profinite completion of on that finite set? If not, what’s a counterexample?
Question 2. Given a compact manifold , does every action of on a finite set extend to a continuous action of the profinite completion of on that finite set? If not, what’s a counterexample?
Question 3. Given a field , is every action of the absolute Galois group of (which is a profinite group) on a finite set continuous? If not, what’s a counterexample?
For questions 1 and 2, isn’t that true for any group G, not just the fundamental groups of a manifold? And moreover, I think of this as the definition of the profinite completion of a group: as an inverse limit over all possible actions of G on finite sets, i.e., all possible homomorphisms G to S_n.
Similarly, for the last one, for any profinite group G acting on a finite set, the stabilizers of any point in the set are finite index which automatically implies that the action is continuous.
Am I missing some subtlety?