## 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?