Hey, I posted a paper to the arXiv and forgot to blog about it! The paper is called “Superstrong approximation for monodromy groups” and it roughly represents the lectures I gave at the MSRI workshop last February on “Thin Groups and Superstrong Approximation.” Hey, as I write this I see that MSRI has put video of these lectures online:
But the survey paper has more idle speculation in it than the lectures, and fewer “um”s, so I recommend text over video in this case! I mean, if you like idle speculation. But if you don’t, would you be reading this blog?
I’m going to recount one of the idle speculations here, but first:
What is superstrong approximation?
Let’s say you have a graph on N vertices, regular of degree d. One basic thing you want to know about the graph is what the connected components are, or at least how many there are. That seems like a combinatorial question, and it is, but in a sense it is also a spectral question: the random walk on the graph, thought of as an operator T on the space of functions on the graph, is going to have eigenvalues between [1,-1], and the mutiplicity of 1 is precisely the number of components; the eigenspace consists of the locally constant functions which are constant on connected components.
So being connected means that the second-largest eigenvalue of T is strictly less than 1. And so you might say a graph is superconnected (with respect to some positive constant x) if the second-largest eigenvalue is at most 1-x. But we don’t say “superconnected” because we already have a word for this notion; we say the graph has a spectral gap of size x. Now of course any connected graph has a spectral gap! But the point is always to talk about families of graphs, typically with d fixed and N growing; we say the family has a spectral gap if, for some positive x, each graph in the family has a spectral gap of size at least x. (Such a family is also called an expander family, because the random walks on those graphs tend to bust out of any fixed-size region very quickly; the relation between this point of view and the spectral one would be a whole nother post.)
When does life hand you a family of graphs? OK, here’s the situation — let’s say you’ve got d matrices in SL_n(Z), or some other arithmetic group. For every prime p, your matrices project to d elements in SL_n(Z/pZ), which produce a Cayley graph X_p, and X_p is connected just when those elements generate SL_n(Z/pZ). If your original matrices generate SL_n(Z), their reductions mod p generate SL_n(Z/pZ); this is just the (not totally obvious!) fact that SL_n(Z) surjects onto SL_n(Z/pZ). But more is true; it turns out that if the group Gamma generated by your matrices is Zariski-dense in SL_n, this is already enough to guarantee that X_p is connected for almost all p. This statement is called strong approximation for Gamma.
But why stop there — we can ask not only whether X_p is connected, but whether it is superconnected! That is: does the family of graphs X_p have a spectral gap? If so, we say Gamma has superstrong approximation, which is now seen to be a kind of quantitative strengthening of strong approximation.
We know much more than we did five years ago about which groups have superstrong approximation, and what the applications are when this is so. Sarnak’s paper from the same conference provides a good overview.
Idle speculation: superstrong approximation for Galois groups
Another way to express superstrong approximation is to say that Gamma has property tau with respect to the congruence quotients SL_n(Z/pZ).
In the survey paper, I wonder whether there is some way to talk about superstrong approximation for Galois groups with bounded ramification. For instance; let G be the Galois group of the maximal extension of Q which is tamely ramified everywhere, and unramified away from 2,3,5, and 7. OK, that’s some profinite group. I don’t know much about it. By Golod-Shafarevich I could prove it was infinite, unless I couldn’t, in which case I would toss in some more ramified primes until I could.
We could ask something like the following. Given any finite quotient Q of G, and any two elements of G whose images generated Q, we get a connected Cayley graph of degree 4 on the elements of Q, by means of those two elements and their inverses. Is there a uniform spectral gap for all those graphs?
I have no real reason to think so. But remark: this would imply immediately that every finite-index subgroup of G has finite abelianization, and that’s true. It would also imply that there are only finitely many n such that G surjects onto S_n, and that might be true. Reader survey for those who’ve read this far: do you think there’s a finite set S of primes such that there are tamely ramified S_n-extensions of Q, for n arbitrarily large, unramified outside S?
Acknowledgment: I was much aided in formulating this question by the comments on the MathOverflow question I asked about it.