We give a Thurston-like definition for laminations on higher Teichmüller spaces associated to a surface and a semi-simple group for or . The case or corresponds to the classical theory of laminations on a hyperbolic surface. Our construction involves positive configurations of points in the affine building. We show that these laminations are parametrized by the tropical points of the spaces and of Fock and Goncharov. Finally, we explain how the space of projective laminations gives a compactification of higher Teichmüller space.
"Higher laminations and affine buildings." Geom. Topol. 20 (3) 1673 - 1735, 2016. https://doi.org/10.2140/gt.2016.20.1673