Higher Laminations and Affine Buildings

Understand

We give a Thurston-like definition for laminations on higher Teichmuller spaces associated to a surface $S$ and a semi-simple group $G$ for $G-SL_m$ and $PGL_m$.

• The case $G=SL_2$ or $PGL_2$ corresponds to the classical theory of laminations.
• Our construction involves positive configurations of points in the affine building.
• We show that these laminations are parameterized by the tropical points of the spaces $\X_{G,S}$ and $\A_{G,S}$ of Fock and Goncharov.