## Breuillard’s ICM talk: uniform expansion, Lehmer’s conjecture, tauhat

Emmanuel Breuillard is in Korea talking at the ICM; here’s his paper, a very beautiful survey of uniformity results for growth in groups, by himself and others, and of the many open questions that remain.

He starts with the following lovely observation, which was apparently in a 2007 paper of his but which I was unaware of.  Suppose you make a maximalist conjecture about uniform growth of finitely generated linear groups.  That is, you postulate the existence of a constant c(d) such that, for any finite subset S of GL_d(C),  you have a lower bound for the growth rate

$\lim |S^n|^{1/n} > c(d)$.

It turns out this implies Lehmer’s conjecture!  Which in case you forgot what that is is a kind of “gap conjecture” for heights of algebraic numbers.  There are algebraic integers of height 0, which is to say that all their conjugates lie on the unit circle; those are the roots of unity.  Lehmer’s conjecture says that if x is an algebraic integer of degree n which is {\em not} a root of unity, it’s height is bounded below by some absolute constant (in fact, most people believe this constant to be about 1.176…, realized by Lehmer’s number.)

What does this question in algebraic number theory have to do with growth in groups?  Here’s the trick; let w be an algebraic integer and consider the subgroup G of the group of affine linear transformations of C (which embeds in GL_2(C)) generated by the two transformations

x -> wx

and

x -> x+1.

If the group G grows very quickly, then there are a lot of different values of g*1 for g in the word ball S^n.  But g*1 is going to be a complex number z expressible as a polynomial in w of bounded degree and bounded coefficients.  If w were actually a root of unity, you can see that this number is sitting in a ball of size growing linearly in n, so the number of possibilities for z grows polynomially in n.  Once w has some larger absolute values, though, the size of the ball containing all possible z grows exponentially with n, and Breuillard shows that the height of z is an upper bound for the number of different z in S^n * 1.  Thus a Lehmer-violating sequence of algebraic numbers gives a uniformity-violating sequence of finitely generated linear groups.

These groups are all solvable, even metabelian; and as Breuillard explains, this is actually the hardest case!  He and his collaborators can prove the uniform growth results for f.g. linear groups without a finite-index solvable subgroup.  Very cool!

One more note:  I am also of course pleased to see that Emmanuel found my slightly out-there speculations about “property tau hat” interesting enough to mention in his paper!  His formulation is more general and nicer than mine, though; I was only thinking about profinite groups, and Emmanuel is surely right to set it up as a question about topologically finitely generated compact groups in general.

## Superstrong approximation for monodromy groups (and Galois groups?)

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.

## Torsion in the homology of arithmetic groups, and an Iwasawa algebra puzzle

Kudos to Nicolas Bergeron, Paul Gunnells, and Akshay Venkatesh for organizing a wonderfully interesting conference at BIRS on torsion on the homology of arithmetic groups.  If you had the bad luck not to be in Banff last week, never fear:  they’ve put in an ultra-fancy new recording/streaming system and you can watch most of the talks online.  The introductory talks by Frank Calegari and Nicolas are a great place to start.

I was raised to think of torsion classes in homology as a terrifying mystery that one dealt with by tensoring with the rational numbers as quickly as possible.  But our knowledge about these things is actually starting to accumulate!

Here’s a puzzle that came up while I was talking to Simon Marshall, whose work makes crucial work of the story about completed cohomology of towers of manifolds that Frank Calegari and Matt Emerton have been steadily telling us.

(remark:  everything below is written off the cuff and no details are checked.)

## Is there a noncommutative Siegel’s Lemma?

Let f be the smallest function satisfying the following:

Suppose given two matrices A and B in SL_3(Z), with all entries at most N.  If there is a word w(A,B,A^{-1},B^{-1}) which vanishes in SL_3(Z), then there is a word w'(A,B,A^{-1},B^{-1}) of length at most f(N) which vanishes in SL_3(Z).

What are the asymptotics of f(N)?

The reason for the title is that, if SL_3(Z) is replaced by Z^n, this is Siegel’s lemma:  if two (or, for that matter, k) vectors in [-N..N]^n are linearly dependent, then there is a linear dependency whose height is polynomial in N.  (Here k and n are constants and N is growing.)

I don’t have any particular need to know this — the question came up in conversation at the very stimulating MSRI Thin Groups workshop just concluded.  Sarnak’s notes are an excellent guide to the topics discussed there.