## FI-modules and representation stability, I

Tom ChurchBenson Farb and I have just posted a new paper, “FI-modules:  a new approach to representation stability,” on the arXiv.  This paper has occupied a big chunk of our attention for about a year, so I’m very pleased to be able to release it!

Here is the gist.  Sometimes life hands you a sequence of vector spaces.  Sometimes these vector spaces even come with maps from one to the next.  And when you are very lucky, those maps become isomorphisms far enough along in the sequence; because at that point you can describe the entire picture with a finite amount of information, all the vector spaces after a certain point being canonically the same.  In this case we typically say we have found a stability result for the sequence.

But sometimes life is not so nice.  Say for instance we study the cohomology groups of configuration spaces of points of n distinct ordered points on some nice manifold M.  As one does.  In fact, let’s fix an index i and a coefficient field k and let V_n be the vector space H^i(Conf^n M, k.)

(In the imaginary world where there are people who memorize every word posted on this blog, those people would remember that I also sometimes use Conf^n M to refer to the space parametrizing unordered n-tuples of distinct points.  But now we are ordered.  This is important.)

For instance, you can let M be the complex plane, in which case we’re just computing the cohomology of the pure braid group.  Or, to put it another way, the cohomology of the hyperplane complement you get by deleting the hyperplanes (x_i-x_j) from C^n.

This cohomology was worked out in full by my emeritus colleagues Peter Orlik and Louis Solomon.  But let’s stick to something much easier; what about the H^1?  That’s just generated by the classes of the hyperplanes we cut out, which form a basis for the cohomology group.  And now you see a problem.  If V_n is H^1(Conf^n C, k), then the sequence {V_n} can’t be stable, because the dimensions of the spaces grow with n; to be precise,

dim V_n = (1/2)n(n-1).

But all isn’t lost.  As Tom and Benson explained last year in their much-discussed 2010 paper, “Representation stability and homological stability,” the right way to proceed is to think of V_n not as a mere vector space but as a representation of the symmetric group on n letters, which acts on Conf^n by permuting the n points.  And as representations, the V_n are in a very real sense all the same!  Each one is

“the representation of the symmetric group given by the action on unordered pairs of distinct letters.”

Of course one has to make precise what one means when one says “V_m and V_n are the same symmetric group representation”, when they are after all representations of different groups.  Church and Farb do exactly this, and show that in many examples (including the pure braid group) some naturally occuring sequences do satisfy their condition, which they call “representation stability.”

So what’s in the new paper?  In a sense, we start from the beginning, defining representation stability in a new way (or rather, defining a new thing and showing that it agrees with the Church-Farb definition in cases of interest.)  And this new definition makes everything much cleaner and dramatically expands the range of examples where we can prove stability.  This post is already a little long, so I think I’ll start a new one with a list of examples at the top.

## The hardest Rush Hour position

It takes 93 moves to solve, per this paper by Collette, Raskin, and Servais.  I tried it and got nowhere.

You can think of the space of all possible configurations of vehicles as, well, a configuration space, not unlike the configuration spaces of disks in a box.  But here there is a bit less topology; the space is just a graph, with two configurations made adjacent if one can be reached from the other by making a single move.  The connected component of configuration space containing the “hardest case” shown here has 24,132 vertices.

I wonder what this graph looks like?   What does the path of the cars look like as you traverse the 93-step path; do most of the cars traverse most of their range?  How many of the possible configurations of the 13 vehicles (constrained to stay in the given rows and columns, and in the same linear order when two share a row or column) are actually contained in this component?  Maybe Matt Kahle knows.  By the way, another Matt Kahle-like fact is that among the list of the hardest configurations are some which are not so dense at all, like this one with only 9 cars.  It looks like it should be easy, but apparently it takes 83 moves to solve!

## 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?”

## The space of unknots and the space of unknotted ropes

Here’s something I didn’t know.  Suppose we consider the space K of “long knots” — embeddings of R into R^3, which send t to (t,0,0) whenever |t| > 1.  By closing up the large |t| ends of the arc, you get a knot in S^3.

The path component of K containing the embedding of R along the x-axis is the space of long unknots.  Hatcher proved, as a consequence of his proof of Smale’s conjecture, that the space of long unknots is contractible!  Hatcher also proved that the space of short unknots is homotopic to the Grassmannian of 2-planes in 4-space.  (I take it this follows from the contractibility of the space of long unknots, but didn’t think about it.)  Moreover, Hatcher proves in the unpublished “Topological Moduli Spaces of Knots” that every connected component of K is a $K(\pi,1)$.  When the component corresponds to a torus knot, the fundamental group is Z: for a hyperbolic knot, it is ZxZ.

How do we get a circle in the space of long knots?  Hatcher makes a lovely “moving bead” argument on p.3 of the linked preprint.  Let S be a “short knot” of the given type, i.e. an embedding of S^1 into S^3.  For each point x on S, draw a small bead B around x; then the complement of B in S^3 looks like R^3, and the segment of the knot outside the bead is a long knot.  Now let the bead slide around the knot.  For each x, you get a long knot in the same isotopy class, and this gives a circle contained in the given connected component of K.

I was thinking about this because of Greg Buck‘s very interesting colloquium yesterday.  Greg is interested in the space of knots with positive thickness — what you might call the space of knotted ropes.  Let’s just think about the unknot.  For any knot K in R^3, we can define the radius of K to be the minimal distance between a point x on K and another point on K which lies on the plane through x perpendicular to the knot.  Let U_r be the space of unknots of radius r.  U_0 is just the space considered by Hatcher above, a Gr(2,4).  As we increase r, the space U_r  gets smaller — at some point, it vanishes entirely.  Buck presented a physical argument that U_r is disconnected for some intermediate values of r:  that is, he passed around a  thick closed rope which couldn’t be untangled, but whose meridian is an unknot.

Beyond that, how does the topology of U_r vary with r?  I have no idea — but what a beautiful question!  It is, of course, very reminiscent of questions about configurations of hard discs in a box discussed here earlier.

Buck’s talk centered on an energy functional $\phi$ on the space of knots, which blows up when the knot gets very close to acquiring a self-intersection (i.e. when the radius gets small.)  You might think of $\phi$ as measuring “distance from the boundary of moduli space.”  You might even hope that $\phi$ would be some kind of Morse function on the components of the space of knots!  Indeed, Hatcher expresses a desire for exactly such a function, which would provide a retract of the infinite-dimensional space of knots of a given isotopy type to some finite-dimensional submanifold of minimal energy whose topology we could hope to understand.  At least in some simple cases, Buck’s energy seems to behave like such a function; he showed us magnificent movies of a crimped, tangled knot flowing along the energy gradient to a handsome, easy to grasp, maximal-radius representative of its isotopy class.  Great!

## Pseudo-Anosov puzzle 2: homology rank and dilatation

In fact, following on what I wrote about the two Farb-Leininger-Margalit theorems below, one might ask the following.  Is there an absolute constant c such that, if f is a pseudo-Anosov mapping class on a genus g surface, and the f-invariant subspace of H_1(S) has dimension at least d, then

log λ(f) >= c (d+1)  / g?

This would “interpolate” between Penner’s theorem (the case d=0) and the F-L-M theorem about Torelli (the case d=2g).

## 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?

## McMullen on dilatation in finite covers

Last year I blogged about a nice paper of Thomas Koberda, which shows that every pseudo-Anosov diffeomorphism of a Riemann surface X acts nontrivially on the homology of some characteristic cover of X with nilpotent Galois group.  (This statement is false with “nilpotent” replaced by “abelian.”)  The paper contains a question which Koberda ascribes to McMullen:

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

That question has now been answered by McMullen himself, in the negative, in a preprint released last month.  In fact, he shows that either λ(f) is detected on the homology of a double cover of Σ, or it is not detected by any finite cover at all!

The supremum of the spectral radius of f on the Σ’ is then an invariant of f, which most of the time is strictly bigger than the spectral radius of f on Σ and strictly smaller than λ(f).  Is this invariant interesting?  Are there any circumstances under which it can be computed?

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## Fritz Grunewald, RIP

Fritz Grunewald died unexpectedly this week, just before his 61st birthday.  I never met him but have always been an admirer of his work, and I’d been meaning to post about his lovely paper with Lubotzky, “Linear representations of the automorphism group of the free group.” I’m sorry it takes such a sad event to spur me to get around to this, but here goes.

Let F_n be the free group of rank n, and Aut(F_n) its automorphism group.  How to understand what this group is like?  A natural approach is to study its representation theory.  But it’s actually not so easy to get a handle on representations of this group.  Aut(F_n) acts on F_n^ab = Z^n, so you get one n-dimensional representation; but what else can you find?

The insight of Grunewald and Lubotzky is to consider the action of Aut(F_n) on the homology of interesting finite-index subgroups of F_n.  Here’s a simple example:  let R be the kernel of a surjection F_n -> Z/2Z.  Then R^ab is a free Z-module of rank 2n – 1, and the -1 eigenspace of R^ab has rank n-1.  Now F_n may not act on R, but some finite-index subgroup H of F_n does (because there are only finitely many homomorphisms F_n -> Z/2Z, and we can take H to be the stabilizer of the one in question.)  So H acquires an action on R^ab; in particular, there is a homomorphism from H to GL_{n-1}(Z).  Grunewald and Lubotzky show that this homomorphism has image of finite index in GL_{n-1}(Z).  In particular, when n = 3, this shows that a finite-index subgroup of Aut(F_3) surjects onto a finite-index subgroup of GL_2(Z).  Thus Aut(F_3) is “large” (it virtually surjects onto a non-abelian free group), and in particular it does not have property T.  Whether Aut(F_n) has property T for n>3 is still, as far as I know, unknown.

Grunewald and Lubotzky construct maps from Aut(F_n) to various arithmetic groups via “Prym constructions” like the one above (with Z/2Z replaced by an arbitrary finite group G), and prove under relatively mild conditions that these maps have image of finite index in some specified arithmetic lattice.  Of course, it is natural to ask what one can learn from this method about interesting subgroups of Aut(F_n), like the mapping class groups of punctured surfaces.  The authors indicate in the introduction that they will return to the question of representations of mapping class groups in a subsequent paper.  I very much hope that Lubotzky and others will continue the story that Prof. Grunewald helped to begin.