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.