## 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?

## 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.