During a recent visit, number theorist Jordan Ellenberg told me about a “time-worn analogy” between
(a) A pseudo-Anosov homeomorphism acting on a surface.
(b) The Frobenius automorphism of a smooth algebraic curve .
Jordan has two very interesting posts on this subject, one on what the dilatation should be in case (b) and a recent one where he discusses the finite field analogue of the following question related to the Virtual Haken Conjecture:
Conjecture: A hyperbolic 3-manifold which fibers over the circle has a finite cover with .
As I noted earlier, this is known when the fiber has genus two, or more broadly if the monodromy is hyperelliptic. Intriguingly, Jordan explains the analogous conjecture in the context of (b) is also known in exactly this case…