Low Dimensional Topology

September 10, 2009

Pseudo-Anosov automorphisms and curves over finite fields

Filed under: Mapping class groups,Number theory — Nathan Dunfield @ 10:15 pm

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 X/\mathbb{F}_q.

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 b_1 > 1.

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…

Leave a Comment »

No comments yet.

RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

Blog at WordPress.com.

%d bloggers like this: