Last year I blogged about a nice paper of Thomas Koberda, which shows that every pseudo-Anosov diffeomorphism of a Riemann surface X acts nontrivially on the homology of some characteristic cover of X with nilpotent Galois group. (This statement is false with “nilpotent” replaced by “abelian.”) The paper contains a question which Koberda ascribes to McMullen:

Is the dilatation λ(f) the supremum of the spectral radii of f on Σ’, as Σ’ ranges over finite etale covers of Σ preserved by f?

That question has now been answered by McMullen himself, in the negative, in a preprint released last month. In fact, he shows that either λ(f) is detected on the homology of a *double* cover of Σ, or it is not detected by any finite cover at all!

The supremum of the spectral radius of f on the Σ’ is then an invariant of f, which most of the time is strictly bigger than the spectral radius of f on Σ and strictly smaller than λ(f). Is this invariant interesting? Are there any circumstances under which it can be computed?

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Here’s one example where the spectral radius of the action on homology increases, at least a little, as one takes finite covers.

In particular, consider the genus 2 mapping class “aDEDcb”. The spectral radius is about 1.539222 which is strictly less than the dilatation which is about 1.722083. (The minimal polynomial of the dilatation is x^6 – 2*x^5 + x^4 – x^3 + x^2 – 2*x + 1, and the singularities of the invariant foliation are of types (3, 3, 4)).

I looked at all covers of degree < 19, and there were a number of where the spectral radius increased. The largest spectral radius observed was 1.6355731299 in a certain degree 7 cover.