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 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.
Somehow I missed Koberda’s paper when it first appeared on the ArXiv, but these are certainly interesting results/questions. On it’s face, detecting the dilation using just nilpotent covers would be surprising, but there’s other things about these groups I find confusing and “just are” (e.g. I’ve don’t know a good heuristic reason why the Lawrence-Kramer representations of the braid groups are faithful.)
Also, to nitpick, I think you really mean the Gassner rather than the Burau representation, since the latter looks only at certain abelian covers whereas the former looks at all of them.
Stupid topologists question: When you say “etale cover” should I read etale as a synonym for “finite”, for “unramified”, or both/neither?
Re Gassner/Burau: my underdetermined notation is at fault. what I had in mind was punctured spheres with the punctures unlabeled, so that the group in question is the braid group, not the pure braid group, so you just have the cyclic abelian covers.
etale: more bad notation; in a topological context I should probably have just said “cover,” and what I mean is “unramified.”