Higher laminations and affine buildings

Le, Ian

doi:10.2140/gt.2016.20.1673

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Higher laminations and affine buildings

Ian Le

Geometry & Topology 20 (2016) 1673–1735

DOI: 10.2140/gt.2016.20.1673

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 $X_{G, S}$ and $A_{G, S}$ of Fock and Goncharov. Finally, we explain how the space of projective laminations gives a compactification of higher Teichmüller space.

Keywords

higher Teichmüller theory, compactifications, tropical points, laminations, buildings, flag variety, affine Grassmannian

Mathematical Subject Classification 2010

Primary: 22E40

References

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

Volume 20, issue 3 (2016)