Higher laminations and affine buildings

Abstract

We give a Thurston-like definition for laminations on higher Teichmüller spaces associated to a surface $S$ and a semi-simple group $G$ for $G={SL}_m$ or ${PGL}_m$. The case $G={SL}_2$ or ${PGL}_2$ 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 ${\mathcal{X}}_{G,S}$ and ${\mathcal{A}}_{G,S}$ of Fock and Goncharov. Finally, we explain how the space of projective laminations gives a compactification of higher Teichmüller space.