Torsion in the homology of 3-manifolds

[The post has been slightly edited to clarify some issues of identification of people, as mentioned in the first comment (24-6-09)].

The topic of this post is obviously rather far from my usual fields of expertise, but since I’ve actually used the large sieve to say something about this topic, and since I was reminded of it by the excellent talk of A. Venkatesh during the Bruggeman conference, here are a few words…

The first thing to say is that whereas the topic may seem a tad abstract and a bit far from good old analytic number theory, this is not really true. In fact, in the cases described by Venkatesh (which corresponds to — separate — joint work with F. Calegari and N. Bergeron, and concern hyperbolic arithmetic 3-manifolds), what is involved can be made extremely concrete. Indeed, to get examples, one can take an imaginary quadratic field K, an integral ideal n in K, and ask about the structure of the finitely-generated abelian group defined as the abelianization

of the congruence subgroup

(using the “Bordeaux” notation for the ring of integers of K; amusingly, it seems the post I once wrote on this subject has disappeared, although I could still find a link to it on Google).

The link with homology is that this abelianization is well-known (by the Hurewicz theorem) to be isomorphic to the first homology group

of the quotient manifold of the hyperbolic three space H₃ by the discrete group:

(speaking of notation, I noticed that Venkatesh behaved like an un-reconstructed right-winger in putting his discrete group on the right-hand side).

One outcome of the ongoing work that Venkatesh discussed (still partly conjectural, and encompassing much larger classes of examples) is the fact that the size of the torsion part of these groups tends to be “as large as possible”. As Venkatesh explained, this means roughly that this torsion group is of order exponential with respect to the norm of the ideal n, which is interpreted as being roughly (up to multiplicative constants) the volume of the manifold. The fact that this is the fastest possible type of growth follows from the interpretation of the size of the homology groups (when finite, at least) as the determinant of integral matrices which have only a bounded number of bounded non-zero entries in each row and column (as n varies). A convincing numerical example was given: the torsion subgroup for an ideal of norm about 500 (if I remember right) for

was roughly of order

What I had done on this torsion topic, as explained in Section 7.6 of my almost famous book on the large sieve (and also, though with some uncorrected typos, in a preliminary short note), was to show how, in the model of random (compact connected orientable) 3-manifolds suggested by N. Dunfield and W. Thurston, something quite similar was happening with high probability.

More precisely, the 3-manifolds of Dunfield and Thurston are obtained as follows: one fixes first an integer g at least 2, then a symmetric set of generators of the mapping class group Γ_g of a closed (topological) surface of genus g, then one forms the random walk on this discrete group (where the generators are chosen uniformly and independently), getting a sequence

of random variables, each of which lives in the mapping class group. Because the latter is a big complicated non-abelian group (as can be seen, for instance, quite naively from the important fact that it has a natural surjection

to an integral symplectic group), this random walk is highly transient and “escapes to infinity” quickly. Now, to get from these surface maps to 3-manifolds, one uses “Heegard splittings”: given any mapping class φ, take two filled “handlebodies” of genus g (which are compact 3-manifolds with boundary, the boundary being a standard surface of genus g), then glue them together by identifying the two boundaries (which are, topologically, the same surface) with the help of the homeomorphism φ. [This procedure was invented as early as 1898 by Heegaard; personally, I have the greatest trouble visualizing it, and this has the effect that I find extremely puzzling the proofs I have seen of the fact that any compact 3-manifold can be obtained in this way, at least for some g, since it is more or less treated as close to trivial once the 3-manifold is triangulated…]

In any case, for a 3-manifold presented as a Heegaard splitting of genus g, the abelianization of the fundamental group of the resulting manifold M_φ turns out to be easy to describe in terms of the mapping class used for the identification of the handlebodies: we have

for some lattice J in Z^2g, where the mapping class acts from the fact that the homology of the boundary surface of genus g is a free group of rank 2g.

This formula suggests that, “generically”, the homology group of a 3-manifold should be finite, because, equally generically, the lattice J and its image under the mapping class should be transverse and generate the whole Z^2g together. This was checked qualitatively by Dunfield and Thurston, and using sieve, it was not hard to provide an exponential decay of the probability that the homology is infinite. Then, the main point, already explained by Dunfield and Thurston, is that after fixing a prime p, the analogue formula

holds for the homology with coefficient in the finite field with p elements, which is also equal to

However, there is now a definite probability, of size roughly 1/p, that this homology group be non-zero (roughly, it is the probability that a determinant modulo p, which is mostly equidistributed, vanish). Because the series of 1/p over primes diverges, it follows that, intuitively, the first homology group of a random 3-manifold (in their sense at least) is typically finite, but is divisible by many primes (and hence is quite large).

Making this quantitative (as I did) is not particularly difficult once the large sieve is properly setup in discrete groups with Property (T) (or, in that case, for those having a quotient with property (T), namely the symplectic group Sp(2g,Z); Andersen has shown that the mapping class group itself does not have this property), and it leads to the conclusion that, with probability exponentially close to 1 as the length of the random walk increases, the first homology group is finite with order divisible by roughly

distinct primes. This implies (by Chebychev-type estimates) a growth which is superpolynomial but potentially less than exponential, namely the size should be at least

for some constant α>0, with probability close to 1. If one could show that the primes dividing the order are not always too small (a property which, by the way, was clearly visible in the example shown by Venkatesh, with some fairly large primes appearing), it would follow that the size should be exponential in k, since this is the typical size of an integer with that many prime factors.

At this point one may object that this parameter k, the length of the random walk, is not a good parameter to measure the 3-manifold M_φk with, because it not canonical at all. In fact, although all (compact connected orientable) 3-manifolds have a Heegaard splitting, its genus g can not be chosen at will (one can not always take g=2, as far as I understand, though apparently this is possible for many 3-manifolds, this being related with the minimal number of generators of the fundamental group). Moreover, for a given genus, there are many choices of generating sets to define the random walk, and even then the random walk might come back a few times to the same 3-manifold with different values of k, etc… This suggested that only the asymptotic behavior (as the length of the walk grows) of things like the average number of coverings of the manifold with a given Galois group, etc, really carried significance (such numbers were computed precisely in the paper of N. Dunfield and W. Thurston).

The talk of Venkatesh reminded me of one possible rough interpretation of k, which I vaguely remembered from the paper of Dunfield and Thurston: it is the content of what they state as Conjecture 2.11 (page 12), which says that, with high probability, the random 3-manifold should be hyperbolic, and that its expected volume should grow linearly with k. And if we interpret k as a succédané of the volume, then an exponential lower bound corresponds exactly to the growth conjectured by Bergeron and Venkatesh, while the almost exponential one obtained from the sieve gives strong evidence for it being, indeed, “generic”. [Their manifolds being of a special type, the generic case does not apply directly, of course.]

Here I would have been happy to give some convincing comments on the status of this conjecture of Dunfield and Thurston. J. Maher has a preprint where he proves the hyperbolicity statement, but the proof depends on the geometrization conjecture of Thurston which was, of course, proved by (the methods of) Perelman. Maher also states that the volume part follows then from his result and a work of Brock and Souto, but a web search only reveals this paper to be “in preparation”; added to the fact that I can not claim to understand even Maher’s proof, I feel more comfortable simply saying that it seems that the statement may well be on the verge of being proved. [Any more informed comments will be extremely welcome!]

I would like to conclude by saying that although this is not my area, I found it to be fairly accessible, at an intuitive level, by virtue of the existence of very good surveys and expositions online, e.g., this book-in-progress of Farb and Margalit, and the paper of Dunfield-Thurston, which is very readable.