Crossratios, Surface Groups, SL(n,R) and C^{1}(S^1)\rtimes Diff (S^1)
  We present results connecting crossratios, representations of surface groups in $SL(n,\mathbb R)$ and in an infinite dimensional group related to the group of diffeomorphisms of the circle. More precisely, we show that representations of a surface group in $SL(n,\mathbb R)$ can be interpreted as crossratios satisfying specific algebraic relations, and we explain that all these representations sit together in a space of representations with values in the infinite dimensional group $C^{1,h}(S^1)\rtimes Diff^{h}(S^1)$. 
F. Labourie