In 2006, Labourie defined a map from a bundle over Teichmuller space to the Hitchin component of the representation variety $Rep(\pi_1(S),PSL(n,R))$, and conjectured that it is a homeomorphism for every $n$ (it was known for $n =2,3$). I will describe some of the background to the Labourie conjecture, and then show that it does not hold for any $n >3$. Having shown that Labourie’s map is more interesting than a mere homeomorphism, I will describe some new questions and conjectures about how it might look.

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Last updated: 30 May 2023