Cross ratios, surface groups, PSL(n,𝐑) and diffeomorphisms of the circle
Publications Mathématiques de l'IHÉS, Volume 106  (2007), p. 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 C 1,h (𝕋)Diff h (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.

@article{PMIHES_2007__106__139_0,
     author = {Labourie, Fran\c cois},
     title = {Cross ratios, surface groups, $PSL(n,\mathbf {R})$ and diffeomorphisms of the circle},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer},
     volume = {106},
     year = {2007},
     pages = {139-213},
     doi = {10.1007/s10240-007-0009-5},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2007__106__139_0}
}
Labourie, François. Cross ratios, surface groups, $PSL(n,\mathbf {R})$ and diffeomorphisms of the circle. Publications Mathématiques de l'IHÉS, Volume 106 (2007) , pp. 139-213. doi : 10.1007/s10240-007-0009-5. http://www.numdam.org/item/PMIHES_2007__106__139_0/

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