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This is an experiment: there is a question I want to mention in an article I'm writing, and I am not sure it's a sensible question, so I will ask it here first, in the hopes that if it's insensible someone will tell me. If, on the other hand, the question is sensible, and someone on MO has something interesting to say about it, all the better!

So here is the question. Let S be a finite set of primes $p_1, \ldots, p_k$. Let G be the Galois group of the maximal totally real extension of Q unramified away from S. We want to ask whether G satisfies a version of property tau. What this means is a little tenuous; just to start with, we don't even know (at least I think we don't) whether G is topologically finitely generated. Here's one possible definition:

(E): There exists a constant c such that, for every finite totally real extension K/Q, and every k-tuple $g_1, .. g_k \in Gal(K/Q)$ such that $g_i$ lies in an inertia group over p_i and the $g_1, .. g_k$ together generate Gal(K/Q), the Cayley graph on Gal(K/Q) with generators $g_1, .. g_k$ has spectral gap at least c.

Is (E) plausible? Is (E) even the right notion of "expanding" for unramified Galois groups?

Update: A discreet friend points out that the question is indeed insensible the way I phrased it, because you could take K/Q to be the real subfield of the cyclotomic extension by p^n-th roots. So instead of demanding that K/Q be totally split at $\infty$, it would be better to fix some other prime $\ell$ outside S and require that K/Q split there; that should re-sensibilize the problem. Alternatively, one could drop any kind of splitting requirement and instead ask that K/Q be at worst tamely ramified at the primes in S; this has the additional advantage of tightening the analogy with the function field case.

Update 2: After further thought, I've decided that my favorite version of this question is to ask whether:

For each k, there exists a constant $c_k$ such that, for any Galois extension K/Q which is unramified outside S and tamely ramified at S, and any k-tuple $(g_1, \ldots, g_k)$ generating $G$, the corresponding Cayley graph has spectral gap at most $c_k$.

I don't know whether this is true but I'm pretty sure I don't know that it's false.

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    $\begingroup$ @Will: I do not understand your comment.The extension you propose is ramified at a lot of primes outside of $S$. If you look at the remark before Theorem 10.2.5 in Neukirch-Schmidt-Wingberg's Cohomology of Number Fields, you see thinks are slightly tricker. Indeed, one can prove that $G_S^\text{ab}$ is indeed topologically finitely generated and your extensions are abelian. $\endgroup$ Sep 22, 2012 at 6:23
  • $\begingroup$ @ JSE: Can you give some references for a non-expert who never heard about Cayley graphs and spectral gaps? What is the tau ($\tau$?) property? $\endgroup$ Sep 22, 2012 at 6:25
  • $\begingroup$ Sorry I was thinking backwards, in terms of extensions ramified only outside of $S$, instead of only at $S$. That was silly. $\endgroup$
    – Will Sawin
    Sep 22, 2012 at 7:00
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    $\begingroup$ @Filippo: What is property tau? ams.org/notices/200506/what-is.pdf $\endgroup$ Sep 22, 2012 at 7:34
  • $\begingroup$ @Filippo: See also Agol's answer to mathoverflow.net/questions/31116/… $\endgroup$ Sep 22, 2012 at 7:37

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