In higher Teichmuller theory we study subsets of the character varieties of surface groups that are higher rank analogs of Teichmuller spaces, e.g. the Hitchin components, the spaces of maximal representations and the other spaces of positive representations.
Fock-Goncharov generalized Thurston’s shear coordinates and Penner’s Lambda-lengths to the Hitchin components, showing that they have a beautiful structure of cluster variety.
We applied a similar strategy to Maximal Representations and we found new coordinates on these spaces that give them a structure of non-commutative cluster varieties, in the sense defined by Berenstein-Rethak. This was joint
work with Guichard, Rogozinnikov and Wienhard.
In a project in progress we are generalizing these coordinates to the other sets of positive representations.