I hope this question is appropriate for the site. I've been looking at the expositions of Bers' proof of the Nielsen-Thurston classification given in Hubbard's Teichmüller Theory and Applications to Geometry, Topology, and Dynamics and in Farb-Margalit's A primer on mapping class groups. I was wondering if it's possible to circumvent the use of hyperbolic geometry in the proof.
More precisely, given $S = S_g$, elements are classified according to their translation distance when acting on Teichmueller space $(\mathcal{T}(S), d_{\mathrm{Teich}})$. Here the translation distance of an element is $D(\varphi) = \inf_{\mathcal{X} \in \mathcal{T}(S)} d_\mathrm{Teich}(\mathcal{X}, \varphi \cdot \mathcal{X}) $.
There are three cases:
- If $D(\varphi) = 0$ and the infimum is realized, $\varphi$ is periodic
- If $D(\varphi) >0 $ and the infimum is realized, $\varphi$ is pseudo Anosov.
- If the infimum is not realized, $\varphi$ is reducible.
The first two cases can be handled using only the "analytic" definition of Teichmueller space, as the space of marked Riemann surface structures on $S$. In the first case, you can show that $\varphi$ is (or rather, has a representative that is) conjugate to an isometry of a Riemann surface structure $X$ on $S$, and since $\mathrm{Aut}(X)$ is finite, you are done. In the second, you show $\varphi$ is conjugate to a Teichmueller map and hence pseudo Anosov.
But in the third case, Mumford's compactness theorem is used in a crucial way to produce a multicurve that is preserved by $\varphi$. This involves looking at the lengths of closed geodesics with respect to the hyperbolic metric on a Riemann surface, and then controlling how much $\varphi$ can distort these lengths.
So, the questions that I have in mind are (the last two I'm interested in not only because of the proof of this particular theorem, but for their own sake):
- Is there any other way of producing an invariant multicurve only using Riemann surface structures rather than hyperbolic structures on $S$?
- Is there some translation for "hyperbolic length of the geodesic representative of the isotopy class of a simple closed curve" to Riemann surface language?
- Is there an analog of Mumford's compactness theorem (describing some family of compact subsets of moduli space that also give an exhaustion of it) in "analytic" terms?
I'm not hoping for a positive answer to question 1, since I've never seen a complex analysis statement that says anything that would imply "this curve should go exactly here". It (and question 2) might also be a bit misguided, since hyperbolic and Riemann surface structures on a surface are so closely related that it wouldn't make sense to separate them. But maybe there is something that can be done. Thanks!
