Higher Laminations and Affine Buildings

TL;DR

This paper extends the classical theory of laminations to higher Teichmüller spaces using affine buildings and positive configurations, providing a new parametrization and compactification method.

Contribution

It introduces a Thurston-like definition for higher laminations on Teichmüller spaces associated with semi-simple groups, generalizing classical laminations.

Findings

Laminations are parameterized by tropical points of certain spaces.

The construction involves positive configurations in affine buildings.

Provides a compactification of higher Teichmüller spaces.

Abstract

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. Finally, we explain how these laminations give a compactification of higher Teichmuller spaces.