Notes on Labourie's cyclic surfaces

In January 2015 a great workshop was organized in Asheville by B. Collier, Q. Li. and A. Sanders. Only young mathematicians were invited, and each one had to give a talk on a research paper that was not his/her. I lectured on a (then) recent preprint by F. Labourie where he discussed cyclic surfaces and the proof of his conjecture in rank 2.

I invite you to read Labourie's preprint for a clearer abstract than I could ever write here. I'll just mention what's cool: Hitchin proved that a connected component (or sometimes 2) of the moduli space of surface representations into split adjoint real group $G$ (now called the Hitchin component) is always contractible. He gave an explicit diffeomorphism of this component with a complex vector space, which is super-cool, but has the drawback to rely on an a priori fixed complex structure on the Riemann surface.

Now, when $G = \text{PSL}(2,\mathbb{R})$, this Hitchin component is just the Teichmüller space. Because of this, fixing a complex structure is rather unsatisfying, because the Teichmüller space has no natural base point. Labourie's conjecture hopes to get rid of this unnatural thing by giving a natural fibration of the Hitchin component onto the Teichmüller space, in such a way that the "natural" complex structure to consider, when working with a Hitchin representation $\rho$, is the image of $\rho$ under this fibration.

The preprint revised here does so when $G$ has at most rank two (there are three such groups: $\text{SL}(3, \mathbb R)$, $\text{Sp}(4, \mathbb R)$ and an exceptional group). The technique is very cool, although not generalizable to higher rank. It relies on a careful study of the Lie algebraic property of the considered groups, that are "small enough" that all the interesting Higgs bundles with values there are cyclic (they take values in the root spaces of simple roots or of the shortest root).

Link: Click here to access the notes

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