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?