Higher laminations and affine buildings

We give a Thurston-like definition for laminations on higher Teichmüller spaces associated to a surface S and a semi-simple group G for G D SL_m or PGL_m. The case G D SL₂ or PGL₂ 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 X_G;S and A_G;S of Fock and Goncharov. Finally, we explain how the space of projective laminations gives a compactification of higher Teichmüller space.