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.)

Suppose given a manifold M whose fundamental group pi maps to SL_2(A), where A is the ring of integers of some p-adic field. A natural class of such guys is provided by hyperbolic 3-manifolds, whose fundamental groups sit inside SL_2(O) for O the ring of integers of a number field with a complex embedding. To make life a little simpler, let’s replace M with a finite cover so that the image of pi lies in a pro-p subgroup G of SL_2(A).

Now let K be the kernel of the map from the pro-p completion of pi to G, and let X be the abelianization of K. What is this thing? It’s a Z_p-module that carries a continuous action of G; so we can think of it as a module for the completed group algebra Lambda = Z_p[[G]]. This module, as you might expect from Iwasawa theory, turns out to be a nice way of packaging the behavior of the homology of covers of M in the tower associated to congruence subgroups of G.

So what can we say about it? It’s not hard to see that it’s finitely generated. But a more wonderful fact is that, in many cases, Calegari and Emerton prove that X is a *torsion* module; this tells you (among other things) that the Betti numbers of covers of M grow more slowly than their volume — indeed, at most like (volume)^a with a bounded away from 1.

But how slowly? That question remains somewhat poorly understood. So let me just write down a relevant algebraic question which makes no reference to manifolds, homology, or arithmetic groups at all.

Let X be a finitely generated torsion module for Lambda = Z_p[[G]], where G is a pro-p p-analytic group of dimension d. (I have in mind SL_2 but the question makes sense in general.) Let G_n be the congruence subgroup of level p^n, and let I_n be its augmentation ideal. Then for each n we can consider

X_n := X / I_n X

which is a finite-rank Z_p-module. An old theorem of Harris ensures that

This is, in the end, where the upper bound on the growth of Betti numbers in the tower of manifolds comes from.

But how sharp is this? In particular:

**Question:** For which X is dim(X_n) bounded *below* by a multiple of ?

Certainly this can happen sometimes; for instance, if H is a 1-dimensional subgroup of G, then Ind_H^G Z_p will have this property. But one wonders whether this is in some sense the only way dim X_n can grow so fast.

Simplest nontrivial example: G = Z_p^2. Then Z_p[[G]] is Z_p[[X,Y]], and we can reduce to the case of a torsion module of the form

Z_p[[X,Y]/f(X,Y)

for some power series f. In this case, I_n is the ideal generated by (1+X)^{p^n} – 1 and (1+Y)^{p^n} – 1. And so we are asking about power series f with many zeroes of the form

with p-power roots of unity. One such power series is f(X,Y) = X, which gives us a torsion module induced from a one-dimensional subgroup, as above. What are the others?

For the statement “This module, as you might expect from Iwasawa theory, turns out to be a nice way of packaging the behavior of the homology of covers of M in the tower associated to congruence subgroups of G.” – do you need to know that pi is a good group to conclude this? Goodness was proved for Bianchi groups by Grunewald, Jaikin-Zapirain, and Zalesskii, and their argument now extends to hyperbolic 3-manifolds in general. http://projecteuclid.org/euclid.dmj/1215032810

I think so, yeah, otherwise things get more complicated — haven’t thought this through carefully.

Dear Ian, I don’t think so. What one is computing here is the “limit of the cohomology groups” of a tower of finite normal covers with Galois groups G_N. If the base comes with a finite CW complex, then the corresponding finite complex of free Z_p modules computing cohomology pulls back (at finite level) to a complex of free Z_p[G_N]-modules of the same rank computing the cohomology of the cover. If, for example, G_N = G(Z/p^n) for some arithmetic group G, then the limit of the cohomology will formally be the cohomology of the resulting complex of finite L = Z_p[[G(Z_p)]] modules. The property you are referring to tries to link the limit of the cohomology with the cohomology of the limit (as it were), but that’s not relevant here, in fact it is to some extent the opposite of what one wants to do. Indeed, when G(Z) has the congruence subgroup property, there will certainly be interesting classes of H^*(G(Z),F_p) that don’t come from the cohomology of the pro-finite completion, but the corresponding limits will still be nice finitely generated L-modules, which (in some precise way) contain all the information about the cohomology of the corresponding finite index subgroups.

Ah, now I realize that I should have actually *read* the original post. It looks like JSE tried to write down a short cut for the construction of completed cohomology, but instead of reading it, I just inserted (in my mind) the “correct” definition. So Ian’s complaint is justified. Or rather, JSE’s definition is “wrong”: instead of taking the cohomology of the congruence kernel, one should just take the (appropriate) limit of the cohomology groups at finite level…

Well, I did say it was off the cuff…..just out of curiosity, how different is the abelianized congruence kernel from the completed H_1?

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