### 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 $X$, does every action of $\pi_1(X)$ on a finite set extend to a continuous action of the profinite completion of $\pi_1(X)$ on that finite set? If not, what’s a counterexample?

**Question 2.** Given a compact manifold $X$, does every action of $\pi_1(X)$ on a finite set extend to a continuous action of the profinite completion of $\pi_1(X)$ on that finite set? If not, what’s a counterexample?

**Question 3.** Given a field $k$, is every action of the absolute Galois group of $k$ (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?