Cross ratios, surface groups, PSL(n,𝐑) and diffeomorphisms of the circle
Publications Mathématiques de l'IHÉS, Tome 106 (2007), pp. 139-213.

This article relates representations of surface groups to cross ratios. We first identify a connected component of the space of representations into PSL(n,𝐑) - known as the n-Hitchin component - to a subset of the set of cross ratios on the boundary at infinity of the group. Similarly, we study some representations into C1,h(𝕋)Diffh(𝕋) associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety contains all n-Hitchin components as well as the set of negatively curved metrics on the surface.

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Labourie, François. Cross ratios, surface groups, $PSL(n,\mathbf {R})$ and diffeomorphisms of the circle. Publications Mathématiques de l'IHÉS, Tome 106 (2007), pp. 139-213. doi : 10.1007/s10240-007-0009-5. http://www.numdam.org/articles/10.1007/s10240-007-0009-5/

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