Tag Archives: hain

Homology of the Torelli group and negative-dimensional vector spaces

OK, not really.  You know and I know there’s no such thing as a negative-dimensional vector space.

And yet…

The Torelli group T_g is a subject of hot interest to mapping class groups people — it’s the kernel of the natural surjection from the mapping class group Γ_g to Sp_{2g}(Z).  You can think of it as “the part of the mapping class group that arithmetic lattices can’t see,” or at least can’t see very well, and as such it is somewhat intimidating.  We know very little about it, even in small genera.  One thing we do know is that for g at least 3 the Torelli group is finitely generated; this is a theorem of Johnson, and a recent paper by Andy Putman provides a small generating set.  So H_1(T_g,Q) is finite-dimensional.  (From now on all cohomology groups will be silently assigned rational coefficients.)

But a charming argument of Akita shows that, in general, T_g has some infinite-dimensional homology groups.  How do we know?  Because if it didn’t, you would be able to compute the integer χ(T_g) from the formula

χ(T_g) =  χ( Γ_g)/ χ(Sp_{2g}(Z)).

But both the numerator and denominator of the right-hand-side are known, and their quotient is not an integer once g is at least 7.  Done!

At the Park City Mathematics Institute session I visited this summer, there was a lot of discussion of what these infinite-dimensional homology groups of Torelli might look like.  We should remember that the outer action of Sp_2g(Z) on Torelli yields an action of Sp_2g(Z) on the homology of Torelli — so one should certainly think of these spaces as representations of Sp_2g(Z), not as naked vector spaces.  In the few cases these groups have been described explicitly, they are induced from finite-dimensional representations of infinite-index subgroups H of Sp_2g(Z).

I just wanted to record the small observation that in cases like this, there’s a reasonably good way to assign a “dimension” to the homology group!  Namely:  suppose G is a discrete group and H a a subgroup, and suppose that both BG and BH are homotopic to finite complexes.  (This is not quite true for G = Sp_2g(Z), but surely you’re willing to spot me a little finite level structure wherever I need it.)  Let W be a finite-dimensional representation of H and let V be the induction of W up to G.

Now if H were finite-index in G you’d have

dim V = [G:H] dim W

or, what’s the same,

dim V = χ(BH)/χ(BG) dim W

But note that the latter formula makes sense even if H is infinite-index in G!  And this allows you to assign a “dimension” to some infinite-dimensional homology groups.

For instance, consider T_2, which is not finitely generated.  By a theorem of Mess, it’s a free group on a countable set of generators; these generators are naturally in bijection with cosets in Sp_4(Z) of a subgroup H containing SL_2(Z) x SL_2(Z) with index 2.  Compute the Euler characteristics of H and Sp_4 and you find that the “dimension” of H_1(T_2) is -5.

And when you ask Akita’s argument about this case, you find that the purported Euler characteristic of T_2 is 6; a perfectly good integer, but not such a great Euler characteristic for a free group to have.  Unless, of course, it’s a free group on -5 generators.

If you want to see this stuff written up a bit (but only a bit) more carefully, here’s a short .pdf version, which also includes a discussion of the hyperelliptic Torelli group in genus 3.

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Hain-Matsumoto, “Galois actions on fundamental groups of curves…”

I recently had occasion to spend some time with Richard Hain and Makoto Matsumoto’s 2005 paper “Galois actions on fundamental groups and the cycle C – C^-,” which I’d always meant to delve into.  It’s really beautiful!  I cannot say I’ve really delved — maybe something more like scratched — but I wanted to share some very interesting things I learned.

Serre proved long ago that the image of the l-adic Galois representation on an elliptic curve E/Q is open in GL_2(Z_l), so long as E doesn’t have CM.  This is a geometric condition on E, which is to say it only depends on the basechange of E to an algebraic closure of Q, or even to C.

What’s the analogue for higher genus curves X?  You might start by asking about the image of the Galois representation G_Q -> GSp_2g(Z_l) attached to the Tate module of the Jacobian of X.  This image lands in GSp_{2g}(Z_l).  Just as with elliptic curves, any extra endomorphisms of Jac(X) may force the image to be much smaller than GSp_{2g}(Z_l).  But the question of whether the image of rho must be open in GSp_2g(Z_l) whenever no “obvious” geometric obstruction forbids it is difficult, and still not completely understood.  (I believe it’s still unknown when g is a multiple of 4…?)  One thing we do know in general, though, is that when X is the generic curve of genus g (that is, the universal curve over the function field Q(M_g) of M_g) the resulting representation

\rho^{univ}: G_{Q(M_g)} \rightarrow GSp_{2g}(\mathbf{Z}_\ell)

is surjective.

Hain and Matsumoto generalize in a different direction.  When X is a curve of genus greater than 1 over a field K, the Galois group of K acts on more than just the Tate modules (or l-adic H_1) of X; it acts on the whole pro-l geometric fundamental group of X, which we denote pi.  So we get a morphism

\rho_{X/K}: G_K \rightarrow Aut(\pi)

What does it mean to ask this representation to have “big image”?

Continue reading

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