The entropy of Frobenius

Since Thurston, we know that among the diffeomorphisms of surfaces the most interesting ones are the pseudo-Anosov diffeomorphisms; these preserve two transverse folations on the surface, stretching one and contracting the other by the same factor.  The factor, usually denoted \lambda, is called the dilatation of the diffeomorphism and its logarithm is called the entropy. It turns out that \lambda, which is evidently a real number greater than 1, is in fact an algebraic integer, the largest eigenvalue of a matrix that in some sense keeps combinatorial track of the action of the diffeomorphism on the surface.  You might think of it as a kind of measure of the “complexity” of the diffeomorphism.  A recent preprint by my colleague Jean-Luc Thiffeault says much about how to compute these dilatations in practice, and especially how to hunt for diffeomorphisms whose dilatation is as small as possible.

A diffeomorphism f on \Sigma induces an (outer) automorphism of \pi_1(\Sigma) which depends only on the isotopy class of f.  In fact, the isotopy classes of diffeomorphisms are precisely the discrete group Out(\pi_1(\Sigma)), otherwise known as the mapping class group of \Sigma.  (To be precise, the mapping class group is the index-2 subgroup coming from orientation-preserving diffeomorphisms.)   For a thorough introduction to this beautiful group, go straight to Benson Farb and Dan Margalit’s textbook, available online.

Anyway, the dilatation of a diffeomorphism depends only on its mapping class; so from here on in, we think of dilatation as a real-valued function on the mapping class group.  The only facts I want to record about \lambda are:

  • If \Sigma' \rightarrow \Sigma is an unbranched cover which is preserved by f, then the absolute values of the eigenvalues of f in its action on H^1(\Sigma',\mathbf{Z}) are lower bounds for \lambda;
  • There exists a branched cover \Sigma' \rightarrow \Sigma preserved by f (in fact, a double cover) such that \lambda is the largest eigenvalue of the action of f onH^1(\Sigma',\mathbf{Z}), denoted the spectral radius of f on \Sigma'.

A slogan among number theorists of a certain stripe is that the Galois group of Q is a kind of mapping class group — because it acts on the etale fundamental groups of algebraic curves over \overline{\mathbf{Q}}, which are, as groups, just the profinite completions of the surface groups.

So one might ask:  can the notion of dilatation be extended to Gal(Q)?

Probably not.  The most naive way this could make sense would be for the dilatations on the mapping class group to satisfy a rather nice profinite interpolation property; for instance, if f and g agreed on some (say) very large characteristic pro-p quotient of \pi_1(\Sigma), you would want \lambda(f) and \lambda(g) to satisfy some p-adic congruence.  Benson Farb and I discussed this a bit and concluded that nothing like this is true.

But one can, at least, give a good candidate for the dilatation of Frobenius.  Let X/F_q be an smooth algebraic curve and consider the eigenvalues of Frobenius acting on H_1(X).  The first thing you notice is that the archimedean absolute values of the eigenvalues are all the same, no matter what X is; so this probably isn’t the place to look.  The interesting valuation is the p-adic one.

The p-adic valuations of the Frobenius eigenvalues really do depend on the curve; they can, for instance, look like 2g copies of \sqrt{p}, or they can look like g copies of 1 and g copies of p, or anything in between — the product, of course, has to be p^g.

It seems reasonable to normalize by dividing all eigenvalues by \sqrt{p}, so that their product is 1 as in the mapping class group case.  Then we can denote the largest (normalized) spectral radius of a curve over F_q to be the largest normalized p-adic eigenvalue of Frobenius; this is at least 1 and at most \sqrt{p}.

Now suppose we ask that the dilatation of Frobenius satisfies the two properties of dilatation I listed above.  Then the dilatation \lambda of Frobenius on X/F_q should be the spectral radius of some branched cover of X; it is thus at most \sqrt{p}.  On the other hand, the spectral radius of any etale cover of X should be a lower bound for \lambda.  Now we use a fact due to Raynaud, which I learned about from Jacob Stix:  every curve X/F_q of genus at least 2 has an etale cover with spectral radius \sqrt{p}.

So I say that \sqrt{p} is the only reasonable value to assign to “the dilatation of Frobenius.”

Now does someone who knows more about p-adic Teichmuller space than me want to comment on whether this number is justifiable from any p-adic dynamical point of view?

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