This article relates representations of surface groups to cross ratios. We first identify a connected component of the space of representations into - 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 associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety contains all -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},
volume = {106},
year = {2007},
pages = {139-213},
doi = {10.1007/s10240-007-0009-5},
language = {en},
url = {http://dml.mathdoc.fr/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, Tome 106 (2007) pp. 139-213. doi : 10.1007/s10240-007-0009-5. http://gdmltest.u-ga.fr/item/PMIHES_2007__106__139_0/
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