Volume 20, issue 3 (2016)

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Higher laminations and affine buildings

Ian Le

Geometry & Topology 20 (2016) 1673–1735
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 ${\mathsc{X}}_{G,S}$ and ${\mathsc{A}}_{G,S}$ of Fock and Goncharov. Finally, we explain how the space of projective laminations gives a compactification of higher Teichmüller space.

Keywords
higher Teichmüller theory, compactifications, tropical points, laminations, buildings, flag variety, affine Grassmannian
Primary: 22E40
Publication
Received: 16 December 2014
Revised: 1 June 2015
Accepted: 8 July 2015
Published: 4 July 2016
Proposed: Benson Farb
Seconded: Danny Calegari, Walter Neumann
Authors
 Ian Le Department of Mathematics University of Chicago 5734 S. University Avenue, Room 208C Chicago, IL 60637 USA