So I learned about this interesting invariant from a colloquium by Burglind Jöricke.

(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 (otherwise known as Z) to (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

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?

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