  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 $C^{1,h}(S^1)\rtimes Diff^{h}(S^1)$.
