Volume 20, issue 3 (2016)

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Higher laminations and affine buildings

Ian Le

Geometry & Topology 20 (2016) 1673–1735

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 = SLm or PGLm. The case G = SL2 or PGL2 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 XG,S and AG,S of Fock and Goncharov. Finally, we explain how the space of projective laminations gives a compactification of higher Teichmüller space.

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higher Teichmüller theory, compactifications, tropical points, laminations, buildings, flag variety, affine Grassmannian
Mathematical Subject Classification 2010
Primary: 22E40
Received: 16 December 2014
Revised: 1 June 2015
Accepted: 8 July 2015
Published: 4 July 2016
Proposed: Benson Farb
Seconded: Danny Calegari, Walter Neumann
Ian Le
Department of Mathematics
University of Chicago
5734 S. University Avenue, Room 208C
Chicago, IL 60637