Is the Dedekind sum really a function on SL_2?

Here’s an idle thought I wanted to record before I forgot it.

The Dedekind sum comes up in a bunch of disparate places; it’s how you keep track of the way half-integral weight forms like the eta function aka discriminant to the one-twelfth transforms under SL_2, it shows up in the topology of modular knots, the alternating sum of continued fraction coefficients, etc.  It has a weird definition which I find it hard to get a feel for.  The Dedekind sum also satsfies Rademacher reciprocity:

$D(a,b;c) + D(b,c;a) + D(c,a;b) = \frac{1}{12abc}(a^2 + b^2 + c^2 - 3abc)$

If that right-hand side looks familiar, it’s because it’s the very same cubic form whose vanishing defines the Markoff numbers!  Here’s a nice way to interpret it.  Suppose A,B,C are matrices with ABC = 1 and

(1/3)Tr A = a

(1/3)Tr B = b

(1/3)Tr C = c

(Why 1/3?  See this post from last month.)

Then

$a^2 + b^2 + c^2 - 3abc = (1/9)(2 + \mathbf{Tr}([A,B]))$

(see e.g. this paper of Bowditch.)

The well-known invariance of the Markoff form under moves like (a,b,c) -> (a,b,ab-c) now “lifts” to the fact that (the conjugacy class of) [A,B] is unchanged by the action of the mapping class group Gamma(0,4) on the set of triples (A,B,C) with ABC=1.

The Dedekind sum can be thought of as a function on such triples:

D(A,B,C) = D((1/3)Tr A, (1/3) Tr B; (1/3) Tr C).

Is there an alternate definition or characterization of D(A,B,C) which makes Rademacher reciprocity

$D(A,B,C) + D(B,C,A) + D(C,A,B) = (1/9)(2 + \mathbf{Tr}([A,B]))$

more manifest?

Shin-Strenner: Pseudo-Anosov mapping classes not arising from Penner’s construction

Balazs Strenner, a Ph.D. student of Richard Kent graduating this year, gave a beautiful talk yesterday in our geometry/topology seminar about his recent paper with Hyunshik Shin.  (He’s at the Institute next year but if you’re looking for a postdoc after that…!)

A long time ago, Robert Penner showed how to produce a whole semigroup M in the mapping class group with the property that all but a specified finite list of elements of M were pseudo-Anosov.  So that’s a good cheap way to generate lots of certified pseudo-Anosovs in the mapping class group.  But of course one asks:  do you get all pA’s as part of some Penner semigroup?  This can’t quite be true, because it turns out that the Penner elements can’t permute singularities of the invariant folation, while arbitrary pA’s can.  But there are only finitely many singularities, so some power of a given pA clearly fixes the singularities.

So does every pA have a power that arises from Penner’s construction?  This is what’s known as Penner’s conjecture.  Or was, because Balazs and Hyunshik have shown that it is falsitty false false false.

When I heard the statement I assumed this was going to be some kind of nonconstructive counting argument — but no, they actually give a way of proving explicitly that a given pA is not in a Penner semigroup.  Here’s how.  Penner’s semigroup M is generated by Dehn twists Q_1, … Q_m, which all happen to preserve a common traintrack, so that there’s actually a representation

$\rho: M \rightarrow GL_n(\mathbf{R})$

such that the dilatation of g is the Perron-Frobenius eigenvalue $\lambda$ of $\rho(g)$.

Now here’s the key observation; there is a quadratic form F on R^n such that F(Q_i x) >= F(x) for all x, with equality only when x is a fixed point of Q_i.  In particular, this shows that if g is an element of M not of the form Q_i^a, and x is an arbitrary vector, then the sequence

$x, g x, g^2 x, \ldots$

can’t have a subsequence converging to x, since

$F(x), F(gx), F(g^2 x), \ldots$

is monotone increasing and thus can’t have a subsequence converging to F(x).

This implies in particular:

g cannot have any eigenvalues on the unit circle.

But now we win!  Because $\rho(g)$ is an integral matrix, so all the Galois conjugates of $\lambda$ must be among its eigenvalues.  In other words, $\lambda$ is an algebraic number none of whose Galois conjugates lie on the unit circle.  But there are lots of pseudo-Anosovs whose dilatations $\lambda$ do have Galois conjugates on the unit circle.  In fact, experiments by Dunfield and Tiozzo seem to show that in a random walk on the braid group, the vast majority of pAs have this property!  And these pAs, which Shin and Strenner call coronal, cannot appear in any Penner semigroup.

Cool!

Anyway, I found the underlying real linear algebra question very appealing.  Two idle questions:

• If M is a submonoid of GL_n(R) we may say a continuous real-valued function F on R^n is M-monotone if F(mx) >= F(x) for all m in M, x in R^n.  The existence of a monotone function for the Penner monoid is the key to Strenner and Shin’s theorem.  But I have little feeling for how it works in general.  Given a finite set of matrices, what are explicit conditions that guarantee the existence of an M-monotone function?  Nonexistence?  (I have a feeling it is roughly equivalent to M containing no element with an eigenvalue on the unit circle, but I’m not sure, and anyway, this is not a checkable condition on the generating matrices…)
• What can we say about the eigenvalues of matrices appearing in the Penner subgroup?  Balazs says he’ll show in a later paper that they can actually get arbitrarily close to the unit circle, which is actually not what I had expected.  He asks:  are those eigenvalues actually dense in the complex plane?

The conformal modulus of a mapping class

(Warning — this post concerns math I don’t know well and is all questions, no answers.)

Suppose you have a holomorphic map from C^* to M_g,n, the moduli space of curves.  Then you get a map on fundamental groups from $\pi_1(\mathbf{C}^*)$ (otherwise known as Z) to $\pi_1(\mathcal{M}_{g,n})$ (otherwise known as the mapping class group) — in other words, you get a mapping class.

But not just any mapping class;  this one, which we’ll call u, is the monodromy of a holomorphic family of marked curves around a degenerate point.  So, for example, the image of u on homology has to be potentially unipotent.  I’m not sure (but I presume others know) which mapping classes u can arise in this way; does some power of u have to be a product of commuting Dehn twists, or is that too much to ask?

In any event, there are lots of mapping classes which you are not going to see.  Let m be your favorite one.  Now you can still represent m by a smooth loop in M_g,n.  And you can deform this loop to be a real-analytic function

$f: \{z: |z| = 1\} \rightarrow \mathcal{M}_{g,n}$

Finally — while you can’t extend f to all of C^*, you can extend it to some annulus with outer radius R and inner radius r.

Definition:  The conformal modulus of a mapping class x is the supremum, over all such f and all annuli, of (1/2 pi) log(R/r).

So you can think of this as some kind of measurement of “how complicated of a path do you have to draw on M_{g,n} in order to represent x?”  The modulus is infinite exactly when the mapping class is represented by a holomorphic degeneration.  In particular, I imagine that a pseudo-Anosov mapping class must have finite conformal modulus.  That is:  positive entropy (aka dilatation) implies finite conformal modulus.   Which leads Jöricke to ask:  what is the relation more generally between conformal modulus and (log of) dilatation?  When (g,n) = (0,3) she has shown that the two are inverse to each other.  In this case, the group is more or less PSL_2(Z) so it’s not so surprising that any two measures of complexity are tightly bound together.

Actually, I should be honest and say that Jöricke raised this only for g = 0, so maybe there’s some reason it’s a bad idea to go beyond braids; but the question still seems to me to make sense.  For that matter, one could even ask the same question with M_g replaced by A_g, right?  What is the conformal modulus of a symplectic matrix which is not potentially unipotent?  Is it always tightly related to the size of the largest eigenvalue?

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.

A primer on mapping class groups

Benson Farb and Dan Margalit have just finished the final version of A Primer on Mapping Class Groups, to appear from Princeton University Press next year.  And it’s available in .pdf at Dan’s web page.  To the extent I know anything about mapping class groups, it’s because of this book!  Highly recommended for anyone interested in finding a way into this very active area of topology, which has heretofore not been so easy to learn about unless you have the luck to sit at the feet of a master.  Is it too much to say I expect the book to become “the Hartshorne of the mapping class group?”

Pseudo-Anosovs with low dilatation: Farb-Leininger-Margalit, and a puzzle

I spent a very enjoyable weekend learning about the dilatation of pseudo-Anosov mapping classes at a workshop organized by Jean-Luc Thiffeault and myself.  The fact that a number theorist and a fluid dynamicist would organize a workshop about an area in low-dimensional topology should indicate, I hope, that the topic is of broad interest!

There are lots of ways to define dilatation, which is a kind of measure of “complexity” of a mapping class.  Here’s the simplest.  Let f be a diffeomorphism from a genus-g Riemann surface S to itself, which is pseudo-Anosov.  Loosely speaking, this means the dynamics of  f are “irreducible” on the surface; for instance, no power of f acts trivially on any subsurface.  (“Most” diffeomorphisms, in any reasonable sense, are pA.)  For any two curves a,b on S, let i(a,b) be the minimal number of intersection points between a and any curve isotopic to b.  (Note that this is typically a lot bigger than the intersection of the homology classes of a and b; the latter measures the number of intersection points counted with sign, which doesn’t change when you isotop the curves.)  It turns out that the quantity

(1/k) log i(f^k(a),b)

approaches a limit as k grows, which strictly exceeds 1;  this limit is called λ(f), the dilatation of f.  It’s invariant under deformation of f; in other words, it depends only on the class of f in the mapping class group of S.  That this limit exists is exciting enough; better still, and indicative of lots of structure I’m passing over in silence, is that λ(f) is an algebraic integer!

(I just remembered that I gave a different description of the dilatation on the blog last year, in connection with an analogy to Galois groups.)

The subject of the conference was pseudo-Anosovs with low dilatation.  The dilatations of pAs in a given genus g are known to form a discrete subset of the interval (1,infinity); thus it makes sense to ask what the smallest dilatation in genus g is.  Lots of progress on this problem has been made in recent years; Joan Birman, Eriko Hironaka, Chia-Yen Tsai, and Ji-Young Ham all talked about results in this vein.  But for general g the answer remains unknown.

A theorem of Penner guarantees that, for any pseudo-Anosov f on a surface of genus g, we have λ(f) > c^(1/g) for some constant c.  So one might call a family f_1, f_2,…. of pAs of varying genera g_1, g_2, …  “low-dilatation” if the quantity λ(f_i)^g_i is bounded.  (One such family, constructed by Hironaka and Eiko Kin, appeared in many of the lectures.)

In this connection, let me advertise the extremely satisfying theorem of Benson Farb, Chris Leininger, and Dan Margalit.  Here’s a natural construction you can do with a pA diffeomorphism f on a surface S.  The diffeo has an invariant foliation which is stretched by f; this foliation has a finite set of singularities.  Remove this to get a punctured surface S^0.  Since the singularities are preserved setwise by f, we have that f restricts to a diffeomorphism of S^0, which is again pA, and which we again call f.  Now we can make a 3-manifold M^0_f by starting with S^0 x [0,1] and gluing S^0  x 0 to S^0 x 1 via f.  By a theorem of Thurston, this will be a hyperbolic 3-manifold; because of the punctures, it’s not compact, but its ends are shaped like tori.

Now here’s the theorem:  suppose f_1, f_2, … is a sequence of pAs which has low dilatation in the sense above.  Then the sequence of 3-manifolds M^0_{f_i} actually consists of only finitely many distinct hyperbolic 3-manifolds.

This has all kinds of marvelous consequences; it tells us that the low-dilatation pAs are in some sense “all alike.”  (For more on the “in some sense” I would need to talk about the Thurston norm and fibered faces and etc. — maybe another post.)  For instance, it immediately implies that in a low-dilatation family of pseudos, the dimension of the subspace of H_1(S_i) fixed by f_i is bounded.

If you’ve read this far, maybe you’d like to see the promised puzzle.  Here it is.  Suppose f_1, f_2, … is a family of pseudos which lie in the Torelli group — that is, f_i acts trivially on H_1(S_i).  Then by the above remark this family can’t be low-dilatation.  Indeed, an earlier theorem of Farb, Leininger, and Margalit tells us that for Torellis we have an absolute lower bound

λ(f) > c

where the constant doesn’t depend on g.

Puzzle: Suppose f_1, f_2, … is a sequence of pseudos in Torelli which has bounded dilatation; this is as strong a notion of “low-dilatation family” as one can ask for.  Is there a “structure theorem” for f_1, f_2, …. as in the general case?  I.E., is there any “closed-form description” of this family?

Anabelian puzzle 1: lifting Galois representations from symplectic groups to mapping class groups

This is an experimental post:  I’m going to put up something I’ve thought about only vaguely and partially, with the idea that it might be an interesting thing to discuss with people at the upcoming workshop on anabelian geometry at the Newton Institute.  Please forgive (or, better, correct) any and all mistakes.

Here’s an old puzzle of Oort, which I’ve mentioned here previously:

Does there exist, for every g > 3, an abelian variety over Qbar not isogenous to the Jacobian of any smooth genus-g curve?

Here’s one way you could imagine trying to construct an example.  Given a g-dimensional abelian variety A over a number field K, the action of the absolute Galois group G_K on the l-adic Tate module of A provides a Galois representation $\rho: G_K \rightarrow Sp_{2g}(Z_\ell)$.

Now suppose A is in fact the Jacobian of a smooth genus g curve X/K.  Then $\rho$ lifts to a representation $\tilde{\rho}: G_K \rightarrow G$, where G is the automorphism group of the pro-l geometric fundamental group of X.  (In fact, this is the case even if A is only isogenous to a Jacobian, as long as the degree of the isogeny is prime to l.)

So you can ask: are there abelian varieties A such that $\rho$ doesn’t lift from the symplectic group to G?  More:  such that the restriction $\rho | G_L$ doesn’t lift to G, for any finite extension L/K?  This seems quite difficult; it means you have to find an obstruction which isn’t torsion.

My further thoughts on this are even more disorganized and I’ll keep them to myself.  Oh, except I should say:  what makes this puzzle “anabelian” is that it has something to do with the notion that a section of the map

$\pi_1^{et}(M_g/K) \rightarrow G_K$

ought to come from a point of M_g(K).

Yes, there is an anabelian puzzle 2, which I’ll try to post in the next couple of days.

Koberda on dilatation and finite nilpotent covers

One reason dilatation was on my mind was thanks to a very interesting recent paper by Thomas Koberda, a Ph.D. student of Curt McMullen at Harvard.

Recall from the previous post that if f is a pseudo-Anosov mapping class on a surface Σ, there is an invariant λ of f called the dilatation, which measures the “complexity” of f; it is a real algebraic number greater than 1.  By the spectral radius of f we mean the largest absolute value of an eigenvalue of the linear automorphism of $H_1(\Sigma,\mathbf{R})$ induced by f.  Then the spectral radius of f is a lower bound for λ(f), and in fact so is the spectral radius of f on any finite etale cover of Σ preserved by f.

This naturally leads to the following question, which appears as Question 1.2 in Koberda’s paper:

Is λ(f) the supremum of the spectral radii of f on Σ’, as Σ’ ranges over finite etale covers of Σ preserved by f?

It’s easiest to think about variation in spectral radius when Σ’ ranges over abelian covers.  In this case, it turns out that the spectral radii are very far from determining the dilatation.  When Σ is a punctured sphere, for instance, a remark in a paper of Band and Boyland implies that the supremum of the spectral radii over finite abelian covers is strictly smaller than λ(f), except for the rare cases where the dilatation is realized on the double cover branched at the punctures.   It gets worse:  there are pseudo-Anosov mapping classes which act trivially on the homology of every finite abelian cover of Σ, so that the supremum can be 1!  (For punctured spheres, this is equivalent to the statement that the Burau representation isn’t faithful.)  Koberda shows that this unpleasant state of affairs is remedied by passing to a slightly larger class of finite covers:

Theorem (Koberda) If f is a pseudo-Anosov mapping class, there is a finite nilpotent etale cover of Σ preserved by f on whose homology f acts nontrivially.

Furthermore, Koberda gets a very nice purely homological version of the Nielsen-Thurston classification of diffeomorphisms (his Theorem 1.4,) and dares to ask whether the dilatation might actually be the supremum of the spectral radius over nilpotent covers.  I have to admit I would find that pretty surprising!  But I don’t have a good reason for that feeling.

F_1, buildings, the braid group, GL_n(F_1[t,1/t])

It used to be you had to talk about “the field with one element” very quietly, and only among people whose opinion of you was secure. The reason, of course, is that there is no field with one element. Which doesn’t stop people from saying “But if there _were_ a field with one element, what would it be like?”

Nowadays all kinds of people are musing about this odd question in the bright light of day, and no one finds it kooky. John Baez covered the basics in a 2007 issue of This Week’s Finds. And as of a few weeks ago the field with one element has its own blog, “Ceci N’est Pas Un Corps.”

From a recent post on CNPUC, I learned the interesting fact that the braid group on n strands can be thought of as GL_n(F_1[t]).

So here’s a question: what is GL_n(F_1[t,1/t])?